Impact of turbulence on flying insects in tethered and free flight: High-resolution numerical experiments

Flapping insects are remarkably agile fliers, adapted to a highly turbulent environment. We present a series of high-resolution numerical simulations of a bumblebee interacting with turbulent inflow. We consider both tethered and free flight, the latter with all six degrees of freedom coupled to the Navier-Stokes equations. To this end, we vary the characteristics of the turbulent inflow, either changing the turbulence intensity or the spectral distribution of turbulent kinetic energy. Active control is excluded in order to quantify the passive response real animals exhibit during their reaction time delay, before the wing beat can be adapted. Modifying the turbulence intensity shows no significant impact on the cycle-averaged aerodynamical forces, moments, and power, compared to laminar inflow conditions. The fluctuations of aerodynamic observables, however, significantly grow with increasing turbulence intensity. Changing the integral scale of turbulent perturbations, while keeping the turbulence intensity fixed, shows that the fluctuation level of forces and moments is significantly reduced if the integral scale is smaller than the wing length. Our study shows that the scale-dependent energy distribution in the surrounding turbulent flow is a relevant factor conditioning how flying insects control their body orientation.

Figure 1

Setup used in present work. (a) Numerical wind tunnel, with tethered or freely flying insects. Turbulent inflow is imposed in the upstream gray area; a vorticity sponge in the downstream green area damps vortices and thus minimizes their upstream influence. The turbulent inlet imposes a slice of an isotropic turbulence field (b), which has been precomputed in a separate simulation. The turbulence field has been upsampled (c) to match the resolution of the numerical wind tunnel and rescaled preserving dynamic similarity. The gray slice in panel (c) moves through the periodic field ${\underline{u}}^{\prime }(x,y,z)$ at constant speed $u_{\infty }$. In some simulations with larger integral scale $\mathrm{\Lambda }$, four identical insects are computed (d) in one simulation, as explained in Sec. 2b.

Figure 2

Snapshot of a simulation. The virtual bumblebee is tethered in the virtual wind tunnel; the mean flow $u_{\infty }$ in the $x^{\left(g\right)}$ direction compensates for the cruising speed of the insect. Reference frames shown are the global (g) and body fixed (b). The flow field is visualized by the vorticity magnitude at an early instant ($t=0.4T$) before the laminar-turbulent interface reaches the insect. The parameters are $I=0.99$ and $\mathrm{\Lambda }=0.77R$.

Figure 3

Spectra of turbulent kinetic energy for HIT fields averaged over several eddy turnover times. Left: series A with constant $\mathrm{\Lambda }$ and variable $I$; right: series B for $I=0.33$ and $\mathrm{\Lambda }$ variable. Markers are the wave numbers associated with $\mathrm{\Lambda }$, $\lambda$, and $\eta$ (from left to right on each spectrum).

Figure 4

Two HIT fields from the series B, both with identical turbulent kinetic energy (and hence turbulence intensity $I$). Visualized is the isosurface $\left|\underline{\omega }\right|=4\sigma \left(\underline{\omega }\right)$ of vorticity magnitude, where $\sigma$ is the standard deviation. Insets show visual comparison of wing length $R$, integral scale $\mathrm{\Lambda }$, and Taylor microscale $\lambda$.

Figure 5

Visualization of the bumblebees prescribed flapping motion every $0.1T$ time steps.

Figure 6

Tethered flight in turbulence. The integral scale is $\mathrm{\Lambda }=0.77R$ and $I$ varies between 0 and 0.99. Cycle averaged values are represented by box plots. Each of the $N_R$ simulations yields four data points, one for each cycle. In the colored boxes, the line is the median (or two-quantile) of the data, and the limits of the box are the upper and lower quartiles (or four-quantile). The additional vertical line are min/max values excluding outliers, which are shown as individual points with a $◇$ marker. Aerodynamic quantities are forces [(a)–(c), normalized by $mg]$, moments [(d)–(f), normalized by $mgR]$, and aerodynamic power [(g), in W/kg body mass]. The dashed line corresponds to the laminar case ($I=0$) where the insect is aligned with the mean flow.

Figure 7

Tethered flight in turbulence. The turbulence intensity is $I=0.33$ and the integral scale $\mathrm{\Lambda }$ varies between $0.32R$ and $1.54R$. Cycle averaged values are represented by box plots. Each of the $N_R$ simulations yields four data points, one for each cycle. In the colored boxes, the line is the median of the data, and the limits of the box are the upper and lower quartiles. The additional vertical line are min-max values excluding outliers, which are shown as individual points with a $◇$ marker. Aerodynamic quantities are forces [(a)–(c), normalized by $mg$], moments [(d)–(f), normalized by $mgR$], and aerodynamic power [(g), in W/kg body mass]. The dashed line corresponds to the laminar case ($I=0$) where the insect is aligned with the mean flow.

Figure 8

Tethered flight in turbulence. Left column, $I=.33$ and $\mathrm{\Lambda }=1.54R$; right column, $I=.33$ and $\mathrm{\Lambda }=0.32R$. [(a), (b)] An isosurface of the $Q$ criterion is shown to visualize vortical structures. In the case of $\mathrm{\Lambda }=1.54R$ (a), fewer vortices are identified upstream of the insect. [(c), (d)] Corresponding pressure field, where the pressure from the laminar inflow has been subtracted, $\mathrm{\Delta }p=p_{\mathrm{turb}}-p_{\mathrm{lam}}$. Compared to the $\mathrm{\Lambda }=0.32R$ case (d), variations in pressure are of the same order of magnitude but on a larger spatial scale in the $\mathrm{\Lambda }=1.54R$ case (c).

Figure 9

Free flight in turbulent inflow, $I=0.16$ (a), 0.33 (b), 0.60 (c), and 0.99 (d), for fixed $\mathrm{\Lambda }=0.77$. Time evolution for the magnitude of body angular velocity. Individual realizations are shown as thin, gray, dashed lines. Ensemble averaged time evolution is represented by the thick green line and the light green shaded background illustrates the standard deviation. The red dashed line corresponds to the laminar case.

Figure 10

Free flight in turbulent inflow. Shown are the yaw (a), pitch (b), and roll (c) components of the body angular velocity, averaged over the last computed cycle, as a function of the turbulence intensity. Data are represented by a box plot. In the colored boxes, the line is the median of the data, and the limits of the box are the upper and lower quartiles. The additional vertical line correspond to min-max values excluding outliers ($◇$).

Figure 11

Free flight in turbulent inflow, $I=0.33$: [(a)–(c)] $\mathrm{\Lambda }=0.77R$ and [(d)–(f)] $\mathrm{\Lambda }=0.32R$. Time evolution of the three components of the angular velocity vector of the body, in the body system ${\underline{\mathrm{\Omega }}}_b^{\left(b\right)}$. Angular velocities are given in ${}^{\circ }/s$ for easier comparison with results in the literature. Individual realizations are shown as thin gray lines. Ensemble-averaged time evolution is represented by the thick green line, and light green shaded background illustrates the standard deviation. The thick red dashed line corresponds to the laminar case.

Figure 12

Free flight in turbulent inflow. Yaw (a), pitch (b), and roll (c) components of the body angular velocity, averaged over the last computed cycle, as a function of $\mathrm{\Lambda }$. Data are represented by a box plot. In the colored boxes, the line is the median of the data, and the limits of the box are the upper and lower quartiles. The additional vertical line is min-max values excluding outliers ($◇$).

Figure 13

Convergence of forces in the wing thickness.