Ground effect on Undulation and pumping near surfaces

Abstract

Locomotion and fluid pumping near surfaces are ubiquitous in nature, ranging from the slow crawling of snails to the rapid flight of bats. This study categorizes these behaviors based on the Undulation number ($\text{Un}$) and Reynolds number ($Re$). We contrast low $Re$ undulatory propulsion ($\text{Un} > 1$), exemplified by freshwater snails, with high $Re$ flapping propulsion ($\text{Un} < 1$), seen in bats and bees. For snails, we derive lubrication models showing that pumping and swimming speeds scale with $(a/h_0)^2$, a result validated by robotic experiments which also reveal the detrimental effects of surface deformation (high Capillary/Bond ratio). Conversely, for high $Re$ fliers, we examine the ground effect's role in lift enhancement. Biological data from bats (\textit{R. ferrumequinum}) reveal a 2.5-fold increase in lift coefficient during surface-skimming drinking flights, attributed to aerodynamic squeezing effects. Finally, we analyze honeybee fanning, demonstrating how a "jet-vortex" mechanism utilizes ground effect to transport pheromones efficiently against diffusion. These findings provide a unified framework for understanding fluid-structure interactions near boundaries in biological systems.

Figures (6)

• Figure 1: Comparative framework of animal locomotion and pumping near surfaces. The behaviors are categorized by the kinematic mode (Vertical axis: Undulation $\text{Un} > 1$ vs. Flapping $\text{Un} < 1$) and biological function (Horizontal axis: Locomotion vs. Pumping). (Top Row) Low Reynolds number undulation ($\text{Un} > 1$): Represented by freshwater snails near the air-water interface Pandey2023. Whether swimming (left; credit: Drs. David Hu and Brian Chan) or stationary pumping for feeding (right; image from Joo2020), the resulting velocity scales with the square of the wave amplitude relative to the gap height: $\frac{V}{V_{wave}} \sim \frac{3}{2} \left(\frac{a}{h_0}\right)^2$. (Bottom Row) High Reynolds number flapping ($\text{Un} < 1$): Represented by aerial fliers. (Left) A bat flying near the water surface ("drinking on the wing") experiences lift enhancement due to the squeezing effect, scaling with the chord-to-amplitude and chord-to-height ratios: $C_{L0}^{squeeze} \sim \frac{c}{a} \frac{c}{h_0}$ (right; image from Maitra2025) A honeybee fanning near a solid ground generates a pumping flow (vortex street) that follows a similar geometric scaling: $\frac{V_{vortex}}{V_{flap}} \sim \frac{c}{a} \frac{c}{h_0}$ (left; image from Peters2017 and credit to Drs. Jake Peters and Stacey Combes).
• Figure 2: Schematic model of snail locomotion and pumping near a free surface.(a) Locomotion: A snail swims with velocity $V_{swim}$ near an air-water interface Pandey2023. The snail's foot generates a traveling wave with propagation speed $V_{wave}$ and amplitude $a$. The gap height $h(x,t)$ oscillates around a mean film thickness $h_0$. The free surface (air-water interface) is modeled as a stress-free boundary that deforms in response to the pressure field. (b) Pumping: In the stationary pumping mode (where the snail is fixed), the same traveling wave motion ($V_{wave}$) generates a net fluid flow, $V_{pump}$, in the direction opposite to the wave propagation. This mechanism is used by snails for filter-feeding near the water surface.
• Figure 3: Experimental validation of snail pumping efficiency and the effect of surface deformation Pandey2023.(a) Measured flow rate per unit width, $\langle Q \rangle$, plotted against the theoretical lubrication scaling parameter $h_0 V_{wave} (a/h_0)^2$. Symbols represent different gap heights ($H$) and working fluids (Silicone Oil and Glycerin-water). The deviation from the linear prediction (dashed line) at higher values indicates energy loss due to free surface interaction.(b) Normalized pumping efficiency $\frac{V_{pump}}{V_{wave}(a/h_0)^2}$ plotted against the ratio of the Capillary number to the Bond number ($Ca/Bo$). The plot collapses the data from (a) into two distinct regimes: Rigid Limit ($Ca/Bo \ll 1$): When gravity dominates, the surface remains flat, and the pumping efficiency plateaus at the theoretical limit of $3/2$. Deformable Limit ($Ca/Bo \gg 1$): When viscous forces dominate, the surface deforms significantly, reducing efficiency. The data follows a decay scaling of roughly $\frac{1}{6}(Ca/Bo)^{-2}$.
• Figure 4: Theoretical models for aerodynamic ground effect in flapping flight.(a) Potential Flow (Method of Images): A discrete vortex model where the flapping wing is represented by a bound vortex with circulation $-\Gamma(t)$. To satisfy the no-penetration boundary condition at the ground, an image vortex with opposite circulation $+\Gamma(t)$ is placed at an equal depth below the surface. The wing operates at a mean height $h_0$ with reduced frequency $\omega$. (b) Squeezing Flow (Air Cushion Model): A model for the unsteady pressure forces generated when the wing flaps close to the ground. The downward vertical velocity $V(t) = dh/dt$ compresses the air in the gap $h(t)$ , generating a high-velocity horizontal squeeze flow $u(x,t)$. This mechanism creates a pressure cushion that significantly enhances lift for small gap heights ($h \ll c$) and small pitch angles ($\theta$).
• Figure 5: Aerodynamic coefficients and lift decomposition during bat drinking flight Maitra2025. (a) A regime map plotting the drag parameter ($C_{D0}$) against the lift parameter ($C_{L0}$) for both the Big bat (H. pratti) and Small bat (R. ferrumequinum). The data reveals a distinct aerodynamic shift: during straight flight (orange/brown clusters), the lift parameter $C_{L0}^{straight}$ is approximately 2. However, during drinking flight (blue/teal clusters), $C_{L0}^{drinking}$ increases dramatically to approximately 5, representing a 2.5-fold enhancement. (b) Decomposition of the total lift coefficient during drinking flight ($C_{L0}^{Drinking}$). The observed lift is compared to the theoretical prediction from Weissinger’s potential flow model ($C_{L0}^h$, bottom bar segments). The potential flow model significantly underpredicts the total force. The remaining lift is attributed to the unsteady squeezing effect ($C_{L0}^{sq}$, top bar segments), which dominates the force generation, accounting for approximately 60% of the total lift (60.4% for the Big bat and 59.6% for the Small bat).