# Numerical Investigation of Odor-Guided Navigation in Flying Insects: Impact of Turbulence, Wingbeat-Induced Flow, and Schmidt Number on Odor Plume Structures

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Governing Equations and Numerical Method

_{i}are the velocity components, P is the pressure, ρ is the fluid density, and ν is the kinematic viscosity.

#### 2.2. Odor Advection–Diffusion Equation

_{i}is the face-centered velocity obtained from interpolation of the cell-centered velocity u

_{i}.

_{W}, a

_{E}, a

_{N}, a

_{S}, a

_{B}, a

_{F}, a

_{P}, are calculated by discretizing the following diffusion term:

#### 2.3. Wing Model with Torsional Spring

^{2}/(area). As shown in Figure 2a, the kinematics of the wing are defined by stroke angle ($\varphi $) and pitch (θ) angle with zero deviation in the angle with respect to the wing stroke plane. In Figure 2b, the stroke plane angle β is 10° and the inclination angle of the insect body χ is 45°. This stroke plane angle is specifically chosen by performing a series of test cases so that the drag generated during the downstroke approximately balances with the thrust generated during the upstroke for the baseline case.

_{ϕ}= 140° is the stroke amplitude, and f is the flapping frequency.

_{xx}, I

_{yy}, I

_{zz}, I

_{xy}, I

_{xx}, and I

_{xz}are the momentum of inertia of the wing. M

_{aero}, M

_{elastic}, and M

_{gravity}are the momentum due to aerodynamic, elastic, and gravitational forces, respectively. The aerodynamic forces are obtained by integrating the pressure and shear on the surface of the insect.

#### 2.4. Turbulence Generator

_{e}is related to the wavenumber at which the energy is maximum and ${\kappa}_{\eta}={\epsilon}^{1/4}{\nu}^{-3/4}$ is the Kolmogorov wave number (smallest turbulence structures).

_{m}is the amplitude, ${\overrightarrow{\kappa}}_{m}=\left({\kappa}_{x,m},{\kappa}_{y,m},{\kappa}_{z,m}\right)$ is the unit vector of the m

^{th}wave number, ψ

_{m}is the phase angle, ${\overrightarrow{\sigma}}_{m}=\left({\sigma}_{x,m},{\sigma}_{y,m},{\sigma}_{z,m}\right)$ is a unit direction vector.

#### 2.5. Simulation Setup

_{3}(0.57), H

_{2}O (0.66), O

_{2}(0.84), CH

_{4}(0.99), CO

_{2}(1.14), methanol (1.14), ethanol (1.5), and n-octane (3.2). These Schmidt numbers are smaller than most odorant particles in water, including O

_{2}(340), NH

_{3}(360), CO

_{2}(410), ethanol (540), and CH

_{4}(570). A smaller value of Sc means molecular diffusion dominates the odor transport, while a larger value of Sc means momentum diffusion dominates the odor transport. The advance ratio $J={U}_{\infty}/{U}_{tip}$ in this study is 0.315, where the incoming air velocity U

_{∞}is 1.01 m/s, and the mean wing tip velocity U

_{tip}is 3.2 m/s.

_{H}is the odor concentration at the odor source. At the odor source, the normalized odor concentration C* = 1.

#### 2.6. Evaluations of Olfactory Performance

## 3. Results and Discussion

#### 3.1. Effects of Turbulence Intensity

#### 3.2. Effects of Schmidt Number

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the grid. The uppercase letters denote cell-centered variables, the lowercase letters denote the face-centered variables.

**Figure 2.**Schematic of the flapping wing with a torsional spring (

**a**) and wing chord diagram during upstroke and downstroke (

**b**), where θ is the pitch angle, ϕ is the stroke angle, β = 10° is the stroke plane angle, and χ = 45° is the body incline angle.

**Figure 3.**Spectra of synthetic turbulence field using the von Kármán–Pao spectrum as an input (dashed lines) with different velocity fluctuations (

**a**), contour of velocity magnitude of the synthetic turbulence field with velocity fluctuation of 0.25 m/s (

**b**), 0.5 m/s (

**c**), 0.75 m/s (

**d**), and 1 m/s (

**e**). The grid resolution is 128 × 128 × 128 with a grid size of 0.01 m.

**Figure 4.**Case setup for simulating fruit fly odor tracking flight. The simulation was performed in a 320 × 128 × 240 (9.8 million) cell nonuniform Cartesian grid. An odor source with staggered arrangement was placed in front of the fruit fly model to simulate odor plumes. At the odor source the normalized odor concentration C* = 1. On each of the two antennae of the fruit fly, the average value of five sampling points was used to represent the odor detectability.

**Figure 5.**(

**a**) Odor landscape computed by solving the advection–diffusion equation for a passive scalar. The domain of the odor concentration field marked by the white dashed rectangle is used to calculate the normalized odor concentration average value and its standard deviation and (

**b**) streaklines released at the odor source are calculated using a Lagrangian approach.

**Figure 6.**Visualizations of the odor landscape under varying conditions of turbulence intensity (Tu) at the mid-downstroke phase. Streaklines originating from the odor source and odor concentration contours are displayed on the symmetry plane. Panels (

**a1**,

**a2**,

**a3**) correspond to Tu = 0; (

**b1,b2**,

**b3**) to Tu = 0.3; and (

**c1**,

**c2**,

**c3**) to Tu = 0.9. Mean normalized odor concentrations and standard deviations along the x-direction are presented in panels (

**a3**,

**b3**,

**c3**). The domain for these calculations is delineated in Figure 5. Vertical dashed lines in panels (

**a3**,

**b3**,

**c3**) mark the positions of the antennae and tail, with x/c = 0 representing the wing root.

**Figure 7.**Time history of the normalized odor concentration C* for different turbulence intensities (

**a**), odor concentration contour on the fruit fly for Tu = 0 (

**b**), 0.3 (

**c**), and 0.9 (

**d**). The Schmidt number for these cases is Sc = 10. The grey shaded regions denote downstrokes. For the odor concentration distribution on the fruit fly (

**b**–

**d**), the left wing shows the odor concentration on the upper surface of the wing, the right wing shows the odor concentration on the lower surface.

**Figure 8.**Odor concentration contour on the symmetry plane, mean normalized odor concentration and the standard deviation along the x-direction for Sc = 0.5 (

**a1**,

**a2**), Sc = 10 (

**b1**,

**b2**), Sc = 100 (

**c1**,

**c2**) at mid-downstroke. For the average odor concentration and the standard deviation, the calculation domain is shown in Figure 5. The turbulence intensity for these cases is Tu = 0. The vertical dashed lines in (

**a2**,

**b2**,

**c2**) denote the locations of antennae and tail. x/c = 0 is the wing root.

**Figure 9.**Time history of the normalized odor concentration C* for different Schmidt numbers (

**a**); odor concentration contour on the fruit fly for Sc = 0.5 (

**b**), 10 (

**c**), and 100 (

**d**). The turbulence intensity for these cases is Tu = 0. The grey shaded regions denote downstrokes. For the odor concentration distribution on the fruit fly (

**b**–

**d**), the left wing shows the odor concentration on the upper surface of the wing, and the right wing shows the odor concentration on the lower surface.

**Figure 10.**Odor concentration contour on the symmetry plane, mean normalized odor concentration and the standard deviation along the x-direction for Sc = 0.5 (

**a1**,

**a2**), Sc = 10 (

**b1**,

**b2**) and Sc = 100 (

**c1**,

**c2**) at mid-downstroke. For the average odor concentration and the standard deviation, the calculation domain is shown in Figure 5. The turbulence intensity for these cases is Tu = 0.3.

**Figure 11.**Time history of the normalized odor concentration C* for different Schmidt numbers (

**a**); odor concentration contour on the fruit fly for Sc = 0.5 (

**b**), 10 (

**c**), and 100 (

**d**). The turbulence intensity for these cases is Tu = 0.3. The grey shaded regions denote downstrokes. For the odor concentration distribution on the fruit fly (

**b**–

**d**), the left wing shows the odor concentration on the upper surface of the wing, and the right wing shows the odor concentration on the lower surface.

**Table 1.**Comparison of flying speeds, mean wingtip velocities, advance ratios, and Reynolds numbers for forward-flying insects known for their proficiency in odor source tracking for survival.

Insects | Forward Flying Speed (${\mathit{U}}_{\mathit{\infty}}$) | Wing Tip Velocity (${\mathit{U}}_{\mathit{t}\mathit{i}\mathit{p}}$) | Advance Ratio ($\mathit{J}={\mathit{U}}_{\mathit{\infty}}/{\mathit{U}}_{\mathit{t}\mathit{i}\mathit{p}}$) | Reynolds Number (Re) |
---|---|---|---|---|

Fruit fly (current) | 1.01 m/s | 3.2 m/s | 0.315 | 180 |

Bumblebee [29] | 2.5 m/s | 8.75 m/s | 0.286 | 2042 |

Butterfly [39] | 1.14 m/s | 2.79 m/s | 0.41 | 3455 |

Hawkmoth [40] | 2.0 m/s | 4.88 m/s | 0.41 | 7335 |

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**MDPI and ACS Style**

Lei, M.; Willis, M.A.; Schmidt, B.E.; Li, C.
Numerical Investigation of Odor-Guided Navigation in Flying Insects: Impact of Turbulence, Wingbeat-Induced Flow, and Schmidt Number on Odor Plume Structures. *Biomimetics* **2023**, *8*, 593.
https://doi.org/10.3390/biomimetics8080593

**AMA Style**

Lei M, Willis MA, Schmidt BE, Li C.
Numerical Investigation of Odor-Guided Navigation in Flying Insects: Impact of Turbulence, Wingbeat-Induced Flow, and Schmidt Number on Odor Plume Structures. *Biomimetics*. 2023; 8(8):593.
https://doi.org/10.3390/biomimetics8080593

**Chicago/Turabian Style**

Lei, Menglong, Mark A. Willis, Bryan E. Schmidt, and Chengyu Li.
2023. "Numerical Investigation of Odor-Guided Navigation in Flying Insects: Impact of Turbulence, Wingbeat-Induced Flow, and Schmidt Number on Odor Plume Structures" *Biomimetics* 8, no. 8: 593.
https://doi.org/10.3390/biomimetics8080593