Figure 1

Pictures of the obstacles, oriented with flow from left to right: (a) circle, (b) horizontal stadium, (c) square, (d) symmetric aerofoil, and (e) asymmetric aerofoil.Reuse & Permissions

Figure 2

(a) Sketch of the simulation, in this case for a horizontal stadium. A 2D foam is created between two fixed walls and caused to flow in the positive $x$ direction by increasing the area of the region to the left of the dark line of films joining the two walls. The obstacle is created in the center of the channel; each film that touches the obstacle applies an equal force outward in the direction normal to the obstacle, and each bubble applies a pressure force inward at the middle of the shared boundary. The films bunch up at the trailing edge of the obstacle, and the bubble pressures rise at the leading edge due to the flow, leading to drag and lift forces on the obstacle. (b) Example (vertical stadium, area ratio $a_r=6$) of the pressure ($F_P$) and network ($F_T$) contributions to the drag ($x$) and the lift ($y$) as a function of iteration number. The drag forces increase linearly before developing a sawtooth variation, which is linked to a build-up of stress followed by avalanches of T1s in the foam. The horizontal lines show the average drag forces. In this case the pressure and network contributions to the lift are both negligible.Reuse & Permissions

Figure 3

Drag vs. obstacle cross section $H/d_b$. Images are for obstacles with area ratio $a_r=10$ with flow from left to right. (a) Vertical stadium ($a_r=2,3,4,6,8,10$). (b) Horizontal stadium ($a_r=2,4,5,6,8,10,20,30$). (c) Square ($a_r=2,4,6,8,10,L/R=8$). (d) Diamond ($a_r=2,4,6,8,10,L/R=8$).Reuse & Permissions

Figure 4

Drag force on different obstacles. (a) Drag vs. shape at constant cross section $H/d_b\approx 2.1$. The pressure contribution to the drag decreases with the rounding of the leading edge and the network contribution decreases with the rounding of the trailing edge. (b) Drag vs. roundness $R/(L+R)$, interpolating between a square ($R=L/8$) and a circle ($L=0$) with $a_r=10$. The same effect is seen as in (a). (c) Drag vs. obstacle length, measured as $(L+2R)/d_b$, for symmetric aerofoils with $R_1=R_2$ at constant cross section $H/d_b\approx 2.1$. The first point on the left corresponds to a circle ($L=0$), and the second to a horizontal stadium ($L=2R$). The network contribution to the drag decreases slightly with length. (d) Drag vs. radius ratio $R_2/R_1$ for a symmetric aerofoil ($a_r=10,L$ varies). The pressure drag decreases when the leading edge has a smaller radius of curvature.Reuse & Permissions

Figure 5

Lift versus asymmetric aerofoil arc length, scaled by $d_b$. All three of $R_1$, $R_2$, and ${\theta }_1$ are chosen to increase roughly in the same proportion. The lift is always in the negative $y$ direction, and the network contribution is smaller than that due to pressure.Reuse & Permissions

Figure 6

Pressure fields averaged over the duration of the simulation. (a) Square obstacle with $a_r=10$, showing increased pressure upstream of the obstacle and low pressure downstream. (b) Asymmetric aerofoil, with $R_1/\sqrt{d_b}=3$, $R_2/\sqrt{d_b}=0.75$, and ${\theta }_1=\pi /6$, showing low pressure beneath as well as downstream. The increase of pressure upstream is less pronounced, and there is a pressure peak beneath the trailing edge of the aerofoil. (c) Zoom of the typical arrangement of films around the same aerofoil, with bubbles shaded by instantaneous pressure on a scale by which pressure increases with gray intensity. In both representations a region of low pressure is evident beneath the aerofoil—it is this that induces a negative lift—as well as the pressure peak beneath the trailing edge.Reuse & Permissions

Figure 7

(a) An interpolation between a diamond and a circle takes the shape shown, with $a_r=10$. When its sense is flipped relative to the direction of flow, the relative contributions to the pressure and network drag change, while the total drag remains the same: a sharp leading edge and rounded trailing edge reduces the pressure drag and increases the network drag. (b) Instantaneous arrangement of films around a flat-bottomed aerofoil (parameters: $a_r=9$, cross section $H/d_b=1.59$, radius of curvature of leading edge is $R_1/d_b=0.41$, of trailing edge is $R_2/d_b=0.17$, and of upper side is $R_3/d_b=3.46$). The instantaneous values of drag and lift are $F_y^T=-2.90,F_y^P=-2.36,F_x^T=2.03,F_x^P=0.42$. The lift is again negative, both network and pressure contributions are similar, and the total lift is of the same order of magnitude as the total drag.Reuse & Permissions