banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
Self-similar evolution of a body eroding in a fluid flow
Rent this article for
Access full text Article
1. S. Schumm and H. Khan, “Experimental study of channel patterns,” Geol. Soc. Am. Bull. 83, 17551770 (1972).
2. S. Ikeda, G. Parker, and K. Sawai, “Bend theory of river meanders. Part 1. Linear developmentJ. Fluid Mech. 112, 363377 (1981).
3. A. Ward, “Yardangs on Mars: Evidence of recent wind erosion,” J. Geophys. Res., [Solid Earth] 84, 81478166, doi:10.1029/JB084iB14p08147 (1979).
4. A. Ward and R. Greeley, “Evolution of the yardangs at Rogers Lake, California,” Geol. Soc. Am. Bull. 95, 829837 (1984).
5. A. Scheidegger, “A physical theory of the formation of hoodoos,” Pure Appl. Geophys. 41, 101106 (1958).
6. S. Wang, “Coastal hoodoos,” Encyclopedia of Coastal Science (Springer, Netherlands, 2005), pp. 260262.
7. P. K. Shah, “Pathophysiology of coronary thrombosis: Role of plaque rupture and plaque erosion,” Prog. Cardiovasc. Dis. 44, 357368 (2002).
8. H. C. Groen, F. J. Gijsen, A. van der Lugt, M. S. Ferguson, T. S. Hatsukami, A. F. van der Steen, C. Yuan, and J. J. Wentzel, “Plaque rupture in the carotid artery is localized at the high shear stress region: A case report,” Stroke 38, 23792381 (2007).
9. C. Picioreanu, M. C. van Loosdrecht, and J. J. Heijnen, “Two-dimensional model of biofilm detachment caused by internal stress from liquid flow,” Biotechnol. Bioeng. 72, 205218 (2001).
10. G. Nanz and L. E. Camilletti, “Modeling of chemical-mechanical polishing: A review,” IEEE Trans. Semiconduct. Manuf. 8, 382389 (1995).
11. S. Gupta, The Classical Stefan Problem: Basic Concepts, Modelling and Analysis (Elsevier, Amsterdam, 2003).
12. L. Ristroph, M. Moore, S. Childress, M. Shelley, and J. Zhang, “Sculpting of an erodible body by flowing water,” Proc. Natl. Acad. Sci. U.S.A. 109, 1960619609 (2012).
13. H. Helmholtz, “Über diskontinuierliche FlüssigkeitsbewegungenPhilos. Mag. 36, 337346 (1868).
14. G. Kirchhoff, “Zur Theorie freier FlüssigkeitsstrahlenJ. Reine Angew. Math. 70, 289298 (1869).
15. G. Parker and N. Izumi, “Purely erosional cyclic and solitary steps created by flow over a cohesive bed,” J. Fluid Mech. 419, 203238 (2000).
16. P.-Y. Lagrée, “Erosion and sedimentation of a bump in fluvial flow,” C. R. Acad. Sci., Ser. IIB Mech. 328, 869874 (2000).
17. H. Schlichting, Boundary Layer Theory (McGraw-Hill, New York, 1960).
18. C. Pozrikidis, Introduction to Theoretical and Computational Fluid Dynamics (Oxford University Press, New York, 1997).
19. V. V. Sychëv, A. I. Ruban, V. V. Sychev, and G. L. Korolev, Asymptotic Theory of Separated Flows (Cambridge University Press, Cambridge, 1998).
20. J. Hureau, E. Brunon, and P. Legallais, “Ideal free streamline flow over a curved obstacle,” J. Comput. Appl. Math. 72, 193214 (1996).
21. S. Alben, M. Shelley, and J. Zhang, “How flexibility induces streamlining in a two-dimensional flow,” Phys. Fluids 16, 16941713 (2004).
22. G. Batchelor, “A proposal concerning laminar wakes behind bluff bodies at large Reynolds number,” J. Fluid Mech. 1, 388 (1956).
23. G. Parkinson and T. Jandali, “A wake source model for bluff body potential flow,” J. Fluid Mech. 40, 577594 (1970).
24. T. Wu, “Cavity and wake flows,” Annu. Rev. Fluid Mech. 4, 243284 (1972).
25. M. Brillouin, “Les surfaces de glissement d'Helmholtz et la résistance des fluides,” Ann. Chim. Phys. 23, 145230 (1911).
26. H. Villat, “Sur la validité des solutions de certains problèmes d'hydrodynamique,” J. Math. Pures Appl. 10, 231290 (1914).
27. T. V. Kármán, “Über laminaire und turbulente Reibung,” Z. Angew. Math. Mech. 1, 233252 (1921).
28. K. Pohlhausen, “Zur näherungsweisen Integration der Differentialgleichung der laminaren Grenzschicht,” Z. Angew. Math. Mech. 1, 252268 (1921).
29. M. G. Crandall and P.-L. Lions, “Viscosity solutions of Hamilton-Jacobi equations,” Trans. Am. Math. Soc. 277, 142 (1983).
30. J. A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (Cambridge University Press, Cambridge, 1999), Vol. 3.
31. V. M. Falkner and S. W. Skan, “Solutions of the boundary layer equations,” Philos. Mag. 12, 865896 (1931).
32.Here we normalize time by , where cm/hr for the experiments, as opposed to normalizing by tf. The quantity t* estimates the vanishing time in the case of no erosion on the backside of the body, allowing a more faithful comparison between the experimental and simulated front evolution.
33. S. Vogel, “Drag and reconfiguration of broad leaves in high winds,” J. Exp. Bot. 40, 941948 (1989).
34. S. Alben, M. Shelley, and J. Zhang, “Drag reduction through self-similar bending of a flexible body,” Nature (London) 420, 479481 (2002).
35. E. Achenbach, “Distribution of local pressure and skin friction around a circular cylinder in cross-flow up to Re = 5 × 106,” J. Fluid Mech. 34, 625639 (1968).
36. W. P. Graebel, Engineering Fluid Mechanics (Taylor and Francis, New York, 2001).
37. F. Engelund and J. Fredsoe, “Sediment ripples and dunes,” Annu. Rev. Fluid Mech. 14, 1337 (1982).
38. A. Fowler, “Dunes and drumlins,” Geomorphological Fluid Mechanics (Springer, Berlin, 2001), pp. 430454.
39. F. Charru, B. Andreotti, and P. Claudin, “Sand ripples and dunes,” Annu. Rev. Fluid Mech. 45, 469493 (2013).
40. K. Kroy, G. Sauermann, and H. J. Herrmann, “Minimal model for sand dunes,” Phys. Rev. Lett. 88, 054301 (2002).
41. K. Kroy, G. Sauermann, and H. J. Herrmann, “Minimal model for aeolian sand dunes,” Phys. Rev. E 66, 031302 (2002).
42. P.-Y. Lagrée, “A triple deck model of ripple formation and evolution,” Phys. Fluids 15, 2355 (2003).
43. J. T. Hack, “Dynamic equilibrium and landscape evolution,” Theories of Landform Development (State University of New York, 1975), pp. 87102.
44. O. Devauchelle, A. Petroff, A. Lobkovsky, and D. Rothman, “Longitudinal profile of channels cut by springs,” J. Fluid Mech. 667, 3847 (2011).
45. D. Burbank, A. Blythe, J. Putkonen, B. Pratt-Sitaula, E. Gabet, M. Oskin, A. Barros, and T. Ojha, “Decoupling of erosion and precipitation in the Himalayas,” Nature (London) 426, 652655 (2003).
46. A. Matmon, P. Bierman, J. Larsen, S. Southworth, M. Pavich, and M. Caffee, “Temporally and spatially uniform rates of erosion in the southern Appalachian Great Smoky Mountains,” Geology 31, 155158 (2003).
47. R. Camassa, R. M. McLaughlin, M. N. J. Moore, and A. Vaidya, “Brachistochrones in potential flow and the connection to Darwin's theorem,” Phys. Lett. A 372, 67426749 (2008).
48. V. Sychev, “Laminar separation,” Fluid Dyn. 7, 407417 (1972).
49. R. Meyer, “A view of the triple deck,” SIAM J. Appl. Math. 43, 639663 (1983).
50. T. Hou, J. Lowengrub, and M. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312338 (1994).
View: Figures


Image of FIG. 1.

Click to view

FIG. 1.

Experimental study of the erosion of a clay cylinder in flowing water. (a) A water tunnel provides a unidirectional flow, and the cross-section 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 12 ). Interfaces extracted from photographs show the cross-section 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 = 140 ± 2 min.

Image of FIG. 2.

Click to view

FIG. 2.

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 quasi-triangular 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.

Image of FIG. 3.

Click to view

FIG. 3.

High-Re 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 () at a characteristic distance δ(). Fluid shear stress is proportional to the slope of the velocity profile at the surface (darkened).

Image of FIG. 4.

Click to view

FIG. 4.

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 12 ). This formula allows us to estimate the vanishing time, , from the experiments. (b) Log-log plot of the area measurements from the experiment (black) and simulation (gray). Both match the 4/3-power prediction.

Image of FIG. 5.

Click to view

FIG. 5.

Falkner-Skan similarity solutions for flow past wedges. (a) Illustration of the outer and boundary-layer 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 right-angled wedge produces uniform shear stress, which allows the shape to be maintained during erosion.

Image of FIG. 6.

Click to view

FIG. 6.

Erosion of an initially circular body. (a) Interfaces from the simulation at evenly spaced time intervals of 0.06  , with time indicated by the scale bar at right (color). As it shrinks, the body forms a quasi-triangular shape with a wedge-like 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.

Image of FIG. 7.

Click to view

FIG. 7.

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, /, 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 / = 0.64, the body has developed a wedge-like front and the separation point has migrated backwards. (d) For this body, the velocity profiles indicate little variation in the shear stress.

Image of FIG. 8.

Click to view

FIG. 8.

Tendency towards uniform erosion rate. (a) The simulation interface velocity as it varies along the body's arc length at time / = 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 = 4 min (red) and 90 min (blue), corresponding to / = 0.03 and 0.64. The experimental measurement also shows the local erosion rate to become more uniform at later time (adapted from Ristroph 12 ).

Image of FIG. 9.

Click to view

FIG. 9.

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, /*, for the simulation (gray) and experiment (black).

Image of FIG. 10.

Click to view

FIG. 10.

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, (). The drag coefficient changes by less than 15% over the course of shape evolution, indicating that erosion is not a streamlining process.

Image of FIG. 11.

Click to view

FIG. 11.

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 0.06 , 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 / = 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 / = 0.9 leads to further collapse of the shapes.

Image of FIG. 12.

Click to view

FIG. 12.

Measurements of the opening angle, Ψ, against time for initial geometries of a narrow-side ellipse (dotted curve), broad-side ellipse (dashed curve), and circle (solid curve) in the simulation. The opening angle tends to roughly 90° in each case.

Image of FIG. 13.

Click to view

FIG. 13.

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 small-amplitude 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.

Image of FIG. 14.

Click to view

FIG. 14.

The conformal mapping method of Levi-Civita. The outer flow domain, indicated by the gray region in the -plane, is conformal mapped to the upper-half-unit disk of the ζ-plane. The portion of the boundary not in the wake is mapped to the perimeter of the disk, ζ = ξ, with the stagnation point, , mapped to ζ = . The free streamlines are mapped to the diameter of the disk, −1 ⩽ ζ ⩽ 1, with the separation points, ±, mapped to ζ = ±1.


Article metrics loading...



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 self-similar evolution of the body, with an emerging quasi-triangular 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...

This is a required field
Please enter a valid email address
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Self-similar evolution of a body eroding in a fluid flow