2.1 Moth observations

We obtained 49 North and Central American moth specimens with plumose antennae, spanning 37 species and 12 families, after capturing 300 moths from the Nantahala region of North Carolina and 24 additional moths from the University of Connecticut Insect Collection (see acknowledgements). We broke off one antenna from each moth and examined it in a three-dimensional scanning confocal microscope. A moth antenna is hierarchical, consisting of three branched tiers. From largest to smallest, these tiers are the stalk, branch and sensillum, and are coloured green, blue and red respectively in figure 1(b). The stalks we measured have a diameter of $136\pm 36~\unicode[STIX]{x03BC}\text{m}$, comparable to that of a human hair; the branches are $43\pm 14~\unicode[STIX]{x03BC}\text{m}$ in diameter and are barely visible; the sensilla are $5.5\pm 1.4~\unicode[STIX]{x03BC}\text{m}$ and are only visible under microscopy.

We consider two possible situations associated with overall antennal posture. First, consider the stalk (in green) angled into the wind at an angle $\unicode[STIX]{x1D719}<90^{\circ }$. Incoming air at velocity $U$ strikes the stalk and redirects to flow along it. The redirected velocity vector $U^{\prime }$ is shown in figure 1(c), where the air strikes the antenna branches held at angle $\unicode[STIX]{x1D703}$ with respect to the stalk. Previous literature indicates that many insects have muscular control of the tilt angle $\unicode[STIX]{x1D719}$ of their stalk, including hawkmoths (Krishnan et al. Reference Krishnan, Prabhakar, Sudarsan and Sane2012), locusts (Gewecke & Heinzel Reference Gewecke and Heinzel1980), honeybees (Khurana & Sane Reference Khurana and Sane2016) and cockroaches (Nishiyama, Okada & Toh Reference Nishiyama, Okada and Toh2007). Because of this muscular control, the angle of the stalk with respect to the flow is rarely measured. Nevertheless, the stalk is generally held at less than $90^{\circ }$ with respect to the direction of movement, as shown by cockroaches (Nishiyama et al. Reference Nishiyama, Okada and Toh2007) and bees (Sane & Jacobson Reference Sane and Jacobson2006).

If moths wish to maximize their frontal captured surface area, they may hold their antennae vertical to their head, as shown by the top view of a single antenna in figure 1(d). In this case, the air strikes the stalk perpendicularly and the air is ‘cupped’ by the branches extruding at an angle $\unicode[STIX]{x1D6FE}$ into the flow. We do not have data on the frequency with which antennae are held in various positions, but we consider both positions for the sake of completeness. The angles $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D6FE}$ are independent, and therefore provide a complete geometric description for how the branches are attached to the stalk.

Previous investigators have measured by hand antenna geometry for model organisms of plumose antennae such as Bombyx mori (Vogel Reference Vogel1983), and filiform antennae of the family Heliozelidae (Wang et al. Reference Wang, Shang, Hilton, Inthavong, Zhang and Elgar2018) but made few suggestions for geometrical trends across moths. Since the angle $\unicode[STIX]{x1D719}$ of the stalk can be adjusted by muscles at the base of the stalk, we did not measure it. Figure 1(e) shows histograms for the branching angles $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D6FE}$ with a normal distribution curve in black. These angles have clear optima, with averages of $52^{\circ }\pm 12^{\circ }$ and $53^{\circ }\pm 15^{\circ }$ respectively. The standard deviation is 30 % of the average, indicating a wide range and that there are diverse evolutionary pressures on moth angle as we discuss later. Nevertheless, our result suggests when a moth holds its antenna in a forward-facing or perpendicular position, the air strikes the branches at a moderate angle, between $40^{\circ }$ and $60^{\circ }$. This result is surprising because physical intuition suggests the branch angle should be $90^{\circ }$ in order to expose the most surface area to the wind.

We present the dimensions of each tier of the hierarchy in figure 2. Each tier is approximately 10 times larger in length and diameter than the previous tier. The lengths and spacing of all tiers as well as the sensilla diameter are mostly constant across all moth specimens with an allometric exponent of less than 0.1, as shown in supplementary table 3. It is interesting that the diameter increases slightly with body size, possibly due to structural reasons. Estimating each tier as a cylinder, the total surface area of the average plumose antenna is found to be ${\sim}261~\text{cm}^{2}$, the same characteristic surface area as a DVD, with the majority of the surface area distributed among the sensilla, as shown in figure 2(b). If each sensillum, branch, and stalk were to be placed end to end, the total length sums to almost one metre, as indicated by figure 2(c). Thus, a moth antenna packs a surprisingly large area and length, presumably to increase the chance of intercepting odours.

Figure 2.

Dimensions of parts of the moth antenna. For all panels, the stalk is green, the branch is blue and the sensillum is red. (a) Schematic of the moth antenna with the length $L$, diameter $d$ and spacing $S$ of the branches shown. (b) Distribution of surface area across the three levels of hierarchy, the stalk, branch and sensillum. (c) Distribution of length across the three levels of hierarchy, the stalk, branch and sensillum. (d) The relationship between moth mass $M$ (g) and diameter $d$ ($\unicode[STIX]{x03BC}\text{m}$) of the stalk, branch and sensillum. (e) The relationship between moth mass $M$ (g) and length $L$ ($\unicode[STIX]{x03BC}\text{m}$) of the stalk, branch and sensillum. (f) The relationship between moth mass $M$ (g) and spacing $S$ ($\unicode[STIX]{x03BC}\text{m}$) of the branch and sensillum. For panels (d–f), values for the stalk, branch and sensillum are represented by green squares, blue circles, and red triangles, respectively.

Figure 3.

Flow around an angled cylinder. (a) Odour molecules, given by the solid blue circles, flow past an angled cylinder. The dashed blue lines denote two streamlines, marked A and B that are deformed by the influence of the cylinder. The dotted green line indicates the distance $x$ from the cylinder that odour molecules can diffuse onto the cylinder. Once close to the cylinder, odour molecules change their path due to the pressure difference $P_{max}-P_{min}$ between the leading and trailing edges of the cylinders. Blue and green lines are originally aligned with the main axis of the cylinder before encountering the pressure gradient and moving around the cylinder in three-dimensional space. (b–d) Smoke rising to impact cylinders angled at $90^{\circ }$, $45^{\circ }$ and $15^{\circ }$ with respect to the incoming flow. Gravity $g$ is in the direction shown. (e–g) Simulation of flow and streamlines around $90^{\circ }$, $45^{\circ }$ and $10^{\circ }$ angled cylinders to flow respectively. Values range from $0.5~\text{m}~\text{s}^{-1}$ (red) to $0~\text{m}~\text{s}^{-1}$ (blue) with linear colour regression.

Using the dimensions presented in figure 2, we can determine whether neighbouring structures influence each other. A well-established criterion to determine this effect is related to the ‘leakiness’ of the structure first coined by Cheer and Koehl (Cheer & Koehl Reference Cheer and Koehl1987b). For the sensilla level, the spacing to diameter ratio is approximately 4, and the Reynolds number based on the sensillum diameter is 0.2. These parameters suggest a moderately leaky system. On the other hand, the branch level is very close to fully leaky with a spacing to diameter ratio of 3 and Reynolds number of 2. As such, the assumption of looking at a branch as an individual cylinder is much more valid than it would be at the sensillum level. We proceed by conducting smoke visualizations around a scaled-up model of a single branch of the antenna.

2.2 Smoke visualization

We perform experiments using a wooden dowel and incense particles of one micron characteristic diameter. The rising speed yields a Reynolds number of 75 and Péclet number of $10^{9}$. Figure 3(b–d) shows the wooden dowel angled at decreasing angles with respect to the flow ($90^{\circ }$, $45^{\circ }$ and $15^{\circ }$). In figure 3(b), the dowel is perpendicular to the flow, and incoming smoke simply strikes the cylinder and curves around it, spreading as it does so. This phenomenon is comparable to the well-known flow past a bluff body (Munson et al. Reference Munson, Okiishi, Huebsch and Rothmayer2013). However, in figure 3(c,d), the rod is at more acute angles with the flow, a feature which changes the path of the smoke dramatically. Smoke travels down the axis of the rod before curving and going around it. This phenomenon is also shown schematically in figure 3(a): for rods angled at less than $90^{\circ }$ the smoke takes a compromise between the initial flow direction and a path perpendicular to the cylinder. We call this phenomenon ‘the lingering effect’ because the flow spends more time in the vicinity of the rod than in the perpendicular case. Inclining the cylinder also further decreases its frontal cross-sectional area, which in turn reduces the number of incoming streamlines that contact the cylinder. Thus, the lingering effect is strongest for moderate angles of attack.

2.3 Wind tunnel experimentation

The smoke visualization gives us useful intuition on the flows generated, but to obtain quantitative information on particle deposition, we conduct wind tunnel tests. We use small plastic fibres from interdental toothbrushes, shown in figure 4. We tested three fibres at 10 increments of $10^{\circ }$, spanning $0^{\circ }$, $10^{\circ }$, $20^{\circ },\ldots ,90^{\circ }$ with respect to the incoming flow. Small drops were introduced into the air stream using a humidifier containing dyed water, with the most common drop diameter of approximately $10~\unicode[STIX]{x03BC}\text{m}$ (Amador et al. Reference Amador, Mao, DeMercurio, Montero, Clewis, Alexeev and Hu2015). Our low-speed wind tunnel pushes air at speeds of $U=0.5~\text{m}~\text{s}^{-1}$, in which the associated Reynolds number based on fibre diameter $d_{c}$ is $Re=Ud_{c}/\unicode[STIX]{x1D708}=4.7$, which is close to the maximum Reynolds number of 3.9 based on the moth branch diameter and flying speed, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity of air in both cases. The Péclet number is of the order of $10^{9}$, which is $10^{8}$ times larger than for moth olfaction, indicating that particles are predominately deposited by inertial impaction rather than diffusion.

Figure 4.

Drop deposition on model antenna. (a) Rendering of experimental set-up with detail view of fibre in test section. (b) Dyed water agglomerated on antenna mimic at $\unicode[STIX]{x1D703}=0^{\circ }$, $45^{\circ }$ and $90^{\circ }$ from flow direction $U$. (c) Particles collected in thousands per second $\text{d}N/\text{d}t$ after exposed to particle laden flow at varied degrees $\unicode[STIX]{x1D703}$. Black solid points represent mean values of experimental data with error bars of one standard deviation. Red line indicates theoretical prediction at water droplet Péclet number.

The mist agglomerates on the fibre to form larger patches visible under ultraviolet light, as shown in figure 4(b). The area of the fibre coated in fluorescence was converted to a number of drops using the image-processing described in the wind tunnel experiment methods sub-section. There was significantly less deposition at $0^{\circ }$ and $90^{\circ }$, and the greatest deposition between $30^{\circ }$ and $60^{\circ }$. We use image analysis to calculate the number of drops that deposit over the duration of the experiment.

The black filled squares in figure 4(c) show the relation between the rate of drop deposition and the angle of the cylinder. At the zero degree inclination, drops simply blow past the cylinder. Deposition increases with increasing tilt angle, with a maximum collection rate of 9.5 thousand drops captured per second at $60^{\circ }$. The rate of collection at the optimum angle is over 30 % higher than that for cylinders perpendicular to the flow. We would like to point out the similarities in the angle of highest collection rate in figure 4(c) and the most common antenna branch angle in the moth angle histograms in figure 1(e). This correspondence between the moth optima and the collection experiments is surprising because the Péclet numbers for these two cases are different by orders of magnitude. Moreover, as we shall see in our mathematical model, we do not expect an optimal deposition to occur at high Péclet number at any angle other than $90^{\circ }$.

In the deposition experiments, we observe a steep 20 % drop in deposition between $60^{\circ }$ and $70^{\circ }$. Upon double checking the original photographs of the cylinders, we did not find any experimental artefacts that could account for this drop off. Moreover, this drop off is not present in the moth antenna angle histogram.