Expanded scaling relations for locomotion in sloped or cohesive granular beds

Dynamic similarity, while commonly applied in fluid systems, has recently been extended to locomotion problems in granular media. The previous work was limited to locomotors in cohesionless, flat beds of grains under the assumption of a simple frictional fluid rheology. However, many natural circumstances involve beds that are sloped or composed of cohesive (e.g. damp or powdery) grains. Here we derive expanded scaling relations inclusive of these phenomena. To validate the proposed scalings, we perform discrete element method simulations with inclined beds and cohesive grains using rotating “wheels” of various shape families, sizes, and loading conditions in accord with the proposed scaling laws. The data show a good agreement between scaled tests, suggesting the usage of these scalings as a potential design tool for off-road vehicles and extraplanetary rovers and as an analysis tool for biolocomotion in soils.

Figure 1

Example simulation system—case of a rectangular wheel in cohesive grains pictured. A wheel comprising rigidly connected yellow inner particles and blue particles on the edge is given a fixed rotation speed, which causes it to travel to the right. The particles in the bed are shown in green, while the red particles at the bottom are fixed. The pictured domain width is about half of that of the whole simulated domain.

Figure 2

For each of panels (a)–(d), snapshots are shown of rectangular wheels at the same dimensionless time driving in noncohesive particle beds of a particular tilt angle $\theta$. The subfigures correspond to the supposed scale-equivalent test cases A, B, and C. The value of $\theta$ is (a) ${85}^{\circ }$, (b) ${95}^{\circ }$, (c) ${100}^{\circ }$, and (d) ${105}^{\circ }$. The wheels spin clockwise and travel to the right. $\tilde{t}\sim 82,118,81,127$ for the four $\vec{g}$ angles, respectively.

Figure 3

Dimensionless power and traveling velocity comparisons of a set of three rectangle wheels on beds of noncohesive particles, with the angle $\theta$ between the traveling direction and the gravity (a) $\theta ={85}^{\circ }$, (b) $\theta ={95}^{\circ }$, (c) $\theta ={100}^{\circ }$, (d) $\theta ={105}^{\circ }$. Blue curves stand for case A, red for case B, and yellow for case C.

Figure 4

Averaged dimensionless power and traveling velocity comparisons of (a) triangular wheels, (b) rectangular wheels, and (c) hexagonal wheels, at different inclined angles. Case A represented by blue markers, case B red, and case C black.

Figure 5

Snapshots at the same $\tilde{t}$ of the differently shaped/sized wheels in horizontal cohesive particle beds: (a) triangular wheels, (b) rectangular wheels and (c) hexagonal wheels. All tests use the same DEM grains. Different wheel sizes are compared in different columns: small in the first column from the left, medium in the second column, and large in the third column. The wheels spin clockwise and travel to the right. $\tilde{t}\sim 57,61,119$, respectively, for the three shapes.

Figure 6

Dimensionless power and traveling velocity comparisons of (a) a set of three triangle wheels, (b) a set of rectangle wheels, and (c) a set of hexagonal wheels on horizontal beds of cohesive particles. Blue curves stand for small wheels, red curves for medium wheels, and yellow curves for large wheels.