Drag on intruders in granular beds: A boundary layer approach

Rapid Communication

Drag on intruders in granular beds: A boundary layer approach

Jonathan Goldsmith, Hong Guo, Shelby N. Hunt, Mingjiang Tao, and Stephan Koehler

Phys. Rev. E 88, 030201(R) – Published 30 September 2013

Abstract

We performed a parametric study of the drag on vertical intruders with uniform cross sections of different sizes and shapes, from which we developed a semiempirical model. Baffling techniques were used to isolate the contributions of the intruder's different subsurfaces, and we observed size effects and force focusing on edges. We propose a boundary layer approach, whereby the drag is the surface integral of an effective stress over a monolayer of particles contacting the intruder. The stress has a simple lithostatic dependence and is a function of the orientation relative to the intruder's direction of motion. This approach is experimentally verified and is consistent with the semiempirical model.

Received 3 October 2012

DOI:https://doi.org/10.1103/PhysRevE.88.030201

Authors & Affiliations

Jonathan Goldsmith¹, Hong Guo^2,3, Shelby N. Hunt¹, Mingjiang Tao³, and Stephan Koehler^1,*

¹Department of Physics, Worcester Polytechnic Institute, Worcester, Massachusetts 01609, USA
²College of Water Resources and Architectural Engineering, Northwest A&F University, Yangling 712100, People's Republic of China
³Department of Civil and Environmental Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts 01609, USA

^*Present address: Department of Physics, Harvard University, Cambridge Massachusetts 02138, USA; koehler@seas.harvard.edu

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 88, Iss. 3 — September 2013

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Drag force dependence on width at different immersion depths $Z$ for (a) posts and (b) square bars in $d = 5$ mm beads. Curves from Eq. (2) using parameters from Table .Reuse & Permissions
Figure 2
(a) Schematic of a moving square bar and its surrounding boundary layer. Flows of representative particles contacting the intruder in the middle region and the bottom edge are shown. Plane views of the boundary layers surrounding (b) a post and (c) the corner of a square bar.Reuse & Permissions
Figure 3
(a) Schematic of intruder baffling with the boundary layer in contact with the instrumented face. The azimuthal angle of the surface normal $\hat{n}$ is $ϕ$ , and the angle between the surface normal and direction of motion is $ψ$ . (b) Azimuthal dependence of dimensionless components of the normal (red) and tangential (blue) drag forces on side faces (for $ϕ = 0$ ) for three combinations of immersion depth and width: $[w, Z] / cm = [2.17, 6], [0.95, 6], [2.17, 4]$ . Solid curves are Fourier series fitting. Dashed curves show the best fit using the bluff-body drag approach.Reuse & Permissions
Figure 4
Azimuthal dependence of normal and tangential drag on corners of square intruders. (a) Dimensionless drag on bottom corner, effective width $w = 3.71$ cm and height $h = 0.22$ cm at immersion depths $Z = 4, 6$ cm. Height of the integration surface of the stress from the boundary layer approach is $h^{'} = h + d / 2$ . Curves are the dimensionless stresses from Fig. 3b. (b) Drag on the side corner, effective width $w = 3.5$ mm. Colored curves in foreground are predictions using the boundary layer approach. Gray curves in the background show the predicted force where the edge contribution has not been included. The inset is a top view showing the corner in red and its orientation.Reuse & Permissions
Figure 5
Azimuthal dependence of the averaged, normalized drag forces $〈 {\tilde{σ}}^{'} 〉$ for regions of posts immersed to $Z = 6$ cm. The colored markers indicate normalization using the augmented radius $R = (W + d) / 2$ . (a) Wedges from posts with diameters $W = 2.54, 7.62$ cm and solid angles $Θ = 39^{\circ}, 44^{\circ}$ , respectively. (b) Two half-cylinders with diameters $W = 0.63, 2.54$ cm. The gray markers are normalized $F_{x}$ for the smaller half-cylinder using $R = W / 2$ .Reuse & Permissions
Figure 6
Schematic showing the cross section of the decomposition of the stress on the augmented surface for (a) square bar and (b) cylinder into two contributions, which are the stress on the intruder's surface of width $W$ and a cylinder's surface of diameter $d$ .Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics