Physiological and Environmental Impact of Temperature Change on Bumblebee Flight

Abstract

Insects are impressive flyers due to their intricate wing anatomy, exceptional maneuvering abilities, and capacity to fly in harsh conditions. Bees adapt to extreme environmental conditions through thermoregulation, which allows them to lose or gain heat to regulate their body temperature and increase survivability and foraging capability. The temperature of the thorax, which is regulated by heat transfer between the body and the environment must be kept within a specific range to generate enough power to fly. Computational studies examining heat transfer effects on the aerodynamic performance of bees are limited. This study analyzes effects of ambient temperature and heat dissipation from the body on lift and thrust using morphologically accurate models from Bombus pensylvanicus scans. Three-dimensional incompressible Navier-Stokes equations were solved to predict flow around the bee in different environments. The results of the study showed that even though the thoracic activity changes the flow conditions around the wing, the effect of heat transfer given the assumptions made was not found to have a significant effect. Ambient conditions, however, play a crucial role in flight performance. Increasing ambient temperature reduced the pressure surrounding the wings, which led to decreased lift and aerodynamic power production at hovering and forward flight.

1. Introduction

Miniaturizing flying vehicles has become a trend with recent technological progress; hence, insect flight has fascinated researchers due to the complex lift generation mechanism relative to the insect size. Dickinson et al. [1] conducted experiments to investigate the unsteady mechanism of insects using a Drosophila wing. The results showed that the translational mechanism or delayed stall had a vital contribution to lift generation; however, it was inadequate to keep insects aloft. It was noted that rotational circulation and wake capture mechanisms also played an essential role in the aerodynamics of insect flight. Nagai et al. [2] compared experimental and numerical results using a dynamically-scaled bumblebee in hovering and forward flight and concluded that the experimental and numerical results agreed and delayed stall, rotational circulation and wake capture mechanisms were functional; nonetheless, they took place in different ways during the downstroke and upstroke of the wing. Lyu et al. [3] computationally investigated the effect of the Reynolds number on the flight of an assumed rigid flat wing planform of a dronefly (Eristalis tenax) under a very low Reynolds number ($\mathit{Re}<100$). The study showed that tiny insects must have different flapping patterns than bigger insects due to stronger viscous forces that decrease lift and thrust for Re $<$ 70.

To better understand the importance of wing morphology on bee aerodynamic performance, Feaster et al. [4] performed two-dimensional computational simulations to compare a morphologically-accurate model of the bumblebee wing with flat plate and ellipse wings. Their results showed that the wing corrugation is a critical factor because the corrugated wing generates shorter and several leading-edge vortex (LEV) structures than the flat plate and the ellipse during downstroke, thereby leading to elevated lift production of the corrugated wing. Shah et al. [5, 6] considered the effect of wing corrugation in three-dimensional numerical simulations using corrugated and smooth wings of Bombus pensylvanicus in hovering and forward velocities. The results proved that the morphologically-accurate model increased lift by 7–12% and decreased thrust by 49% at the higher forward speeds (2.5 and 4.5 m/s) as there were no major differences during hovering. Flexible wings are able to control and direct flow, thus leading to a high lift-to-drag ratio compared to fixed wings [7]. Tobing et al. [8] computationally analyzed the effect of wing flexibility on the aerodynamics of bumblebees. The results illustrated that the flexible wing provides more stability than the rigid wing due to preventing LEV separation, resulting in obtaining 30% higher lift than the rigid wing.

Bumblebees can fly in cold and warm weather conditions due to their adaptation skills to changing air conditions. Even though they succeed in flying in harsh conditions, their flight performance is limited by the environment [9]. In the study by Heinrich [10, 11], it was reported that the body temperature of bumblebees correlated with ambient temperature and the excessive heat in the thorax due to the high air temperature limited the bee foraging time; thus the bees must transfer heat. Heinrich and Kammer [12] conducted an experimental study on bumblebees about the warming-up of flight muscles and thoracic temperature stabilization of bumblebees. Their results showed that bumblebees must warm up their flight muscles before flying by employing wing-beat until the thoracic temperature is over 30°C and the heat production in the thorax rises with increasing wing-beat frequency. Moreover, the excessive heat in the thorax was transferred to the abdomen and was dissipated by convective heat transfer to stabilize the amount of heat in the thorax. Roberts and Harrison [13] predicted that increasing ambient temperature decreases metabolic rate and the wing-beat frequency, thereby causing lower aerodynamic power production.

Bumblebees inhabit many environments and forage long distances at different temperatures due to their adaptation skill to changing environments by employing heat transfer between body and ambient. How does heat transfer affect their flight and how do they respond to ambient temperature changes? To address these questions, the current study was performed to reveal the thermal impacts on the aerodynamics of insects using computational fluid dynamics (CFD). A morphologically-accurate model of a bumblebee (Bombus pensylvanicus) was used as the computational model. In contrast to experimental studies, computational studies to investigate the effect of thermal properties on bees are limited. The motivation of the current study was to present a computational approach to investigate the thermal response of bumblebee flight in hovering and forward flight.

The study will investigate both the effect of the thoracic activity of bees that results in heat transfer with the surroundings and the impact of ambient temperature changes on the flight performance of bumblebees. Due to the correlation of the body temperature of bees with ambient temperature, different body and ambient temperatures are specified using the experimental data of Heinrich [10, 11]. The effects of heat transfer between the body and ambient temperature on the aerodynamics of bumblebees will be analyzed computationally.

2. Bumblebee

Bees are classified into various groups of insects. There are a variety of bee species that live in a colony with a thousand individuals. Bees inhabit diverse environments ranging from cold arctic regions to humid tropical rain forests to hot deserts due to their superior environmental adaptation [14]. They play a critical role in the pollination of wild plants, flowers and harvests, supporting a balance in Nature as well as economic acquisition [15]. Their survivability, however, highly depends on environmental conditions in their habitat [16]. To tolerate unwanted climate conditions, unique adaptation skills are utilized to maintain their life cycle.

Bumblebees with 300 species are among 20,000 other bee species and their mass ranges from 65 mg for the smallest worker to 830 mg for the largest queen species [17, 18]. They inhabit cold and hot environments and are key species for pollinating wild herbs, crops, and vegetables. Unlike honeybees and other bee species, they forage at lower ambient temperatures and for long hours, carry pollen that is nearly $80\%$ percent of their body weight, and are agile fliers that make complex maneuvers [5, 11].

2.1 Morphology

Morphological characteristics play an important role in understanding the flight mechanics of flying creatures. These parameters are grouped into gross and shape parameters. Gross parameters that are a raw description of insect morphology include a body for an assigned length and added mass for a given wing length, as the shape parameters are the distribution of the gross parameters such as radii of wing area, moment of wing area, and body and wing shapes, etc. [19].

The wing shape and body mass determine the flight style of a flying creature. High aspect-ratio wings allow them to perform complex maneuvers and waste less energy; however, this is not purely explained in terms of aspect-ratio [20]. Other wing parameters like length, wing area, flexibility, body and virtual mass, and wing loading have notable impacts on the efficiency of flight performance of flying animals. The wing flexibility helps the aerodynamic performance and is superior compared with rigid wings. Insects utilize flexible wings, and the flexibility provides delayed stall, increased lift, and decreased drag [7, 21, 22].

The wing morphology can determine the optimal wing motion of bees [23]. The bumblebee wing is comprised of veins and cells (Figure 1(a)), and the shape of the wing differs among bumblebee species. Also, their wings are small and require rapid beats to keep them aloft [5, 18, 24]. Corrugated wings (Figure 1(b)) contribute to the aerodynamic performance and flight stability of the bees [25, 26]. It helps larger LEV formation during the downstroke, thus leading to lift augmentation by extending suction pressure on the wing dorsal side [5].

Figure 1.

The wing of a bumblebee (Bombus pensylvanucus), adopted from Ref. [5].

2.2 Thermoregulation

Bees need sufficient fuel sources and a stable thoracic temperature range to maintain their ability to forage in all weather conditions [10]. There are diverse mechanisms such as hair density, hair color, and heat transfer with environment to increase their survivability due to the fact that different species adapt to their unique habitats [14]. The most significant adaptation is the mechanism of heat loss or gain, named thermoregulation, which tolerates harsh environmental conditions [13]. The contraction of the flight muscles to initiate flight generates power and heat in the thorax (see in Figure 2). The flight muscle temperature must be regulated to maintain required work output from bees [10]. Thermoregulation is an adaptation to severe environmental conditions, increasing the bee survivability and ability to forage long distances efficiently [16].

Figure 2.

Bombus vancouverensis with the morphological accurate computational model of the American bumblebee (Bombus pensylvanucus) on the right half. The bumblebee image on the left half is credited to ref. [16].

The hair coats of bumblebees shown in the left half of Figure 2 provide excellent insulation to limit convective heat loss [27]. Nixon and Hines [16] performed a study to examine the effect of the physical features of bumblebees on their temperature regulation. It is reported that the color and pile length played key roles in the passive heating of bumblebees, increasing their survivability in different environments.

The aerodynamic performance of insects is dependent on the body and ambient temperatures [11]. During the flight, muscles are heated by the wing motion and positively correlate with increasing ambient temperature. However, low thorax temperature restrains power production, as does high temperature, because bees cannot heat the flight muscles at low temperatures. Maintaining thoracic temperatures within a limited range is a vital process to maintain the required power output during flight [10, 28, 29].

Bumblebees require thoracic temperatures within ${30}^{\circ }C-{44}^{\circ }C$ to generate the needed power output to rise from the ground and forage [18]. Regardless of body mass, thoracic temperature controls bee muscle functions. As observed by Heinrich [18], the majority of bumblebees cannot maintain flight below ${10}^{\circ }C$ air temperature, whereas large bees or queens are able to fly at lower ambient temperatures down to $0^{\circ }$ C. To keep the thorax temperature during flight at low and high ambient temperatures, different methods are used to heat up and keep the body warm.

Shivering is a method to raise the temperature level of flight muscles that comprise the larger part of the thorax with a slight contraction of flight muscles before the flight takes place [10, 17]. Most bee species need to warm-up their flight muscles for some tasks including before taking off, pollen collection, cooling and warming up of clusters [30].

During long flight periods, bees transfer heat between their thorax and abdomen, and even ambient to keep the thoracic temperature within optimal conditions. The abdomen plays an important role in the heat balance of bees. Most bees are capable of losing heat by transferring to the abdomen from the thorax [31]. The experiments by Church [27, 32] showed that heat dissipation by convection is a more effective way than heat loss by evaporation of the body water in most small insects. Bumblebees only perform cooling with evaporation when they are compelled to fly at high ambient temperatures [31, 33]. Ambient temperatures can negatively affect bee thoracic activity and their aerodynamic performance. The metabolic rate of bees reduces with increasing air temperature, resulting in elevated thoracic temperature [34, 35]. Moreover, a shift in ambient temperature also leads to decreasing aerodynamic performance due to the density reduction and viscosity increase of the air, requiring bees to increase wing-beat frequency to compensate for the aerodynamic loss [10, 13, 16].

3. Numerical methodology

The flow behavior around the bee is simulated using computational models of the unsteady Navier-Stokes equations. The Navier-Stokes equations are a mathematical representation of Newton’s second law of motion relating forces to fluid momentum. The forces account for surface stresses due to fluid gradients in pressure and fluid viscosity. The set of partial differential equations are known as conservation of momentum and are coupled to an equation for conservation of mass. Together, the solution provides information for how pressure and fluid viscosity affect the fluid motion. When temperature changes are important, an equation for conservation of energy is solved to provide information how the fluid motion is affected by temperature gradients. Thus, the time-dependent motion of flapping bumblebee wings is impacted by the thoracic temperature, air temperature and surrounding air pressure as the bee flies, and can be understood by solving the following equations. The air flow is assumed to be laminar, viscous and incompressible.

The equation for conservation of mass is:

$$\frac{\partial \rho }{\partial t}+\overrightarrow{\nabla }\cdot \left(\rho \overrightarrow{v}\right)=0$$E1

where $\rho$ is density, $t$ is time, and $\overrightarrow{v}$ is the velocity vector.

The momentum conservation equation is:

$$\frac{\partial }{\partial t}\left(\rho \overrightarrow{v}\right)+\overrightarrow{\nabla }\cdot \left(\rho \overrightarrow{v}\overrightarrow{v}\right)=-\overrightarrow{\nabla }p+\overrightarrow{\nabla }\cdot \overline{\tau }+\rho \overrightarrow{g}$$E2

where $p$ is pressure, $\overline{\tau }$ is the viscous stress tensor and $\overrightarrow{g}$ is gravitational vector.

The fluid viscous stress tensor is defined as:

$$\overline{\tau }=\mu \left[\left(\overrightarrow{\nabla }\overrightarrow{v}+\overrightarrow{\nabla }{\overrightarrow{v}}^T\right)-\frac{2}{3}\overrightarrow{\nabla }\cdot \overrightarrow{v}\right]$$E3

where $\mu$ is the dynamic (absolute) viscosity of the fluid.

The energy conservation equation is:

$$\frac{\partial }{\partial t}\left(\mathit{\rho E}\right)+\overrightarrow{\nabla }\cdot \left(\left(\mathit{\rho E}+p\right)\overrightarrow{v}\right)=\overrightarrow{\nabla }\cdot k\nabla T+\overrightarrow{\nabla }\cdot \left(\overline{\tau }\cdot \overrightarrow{v}\right)$$E4

where $E$ is total energy, and $k$ is the conduction coefficient term.

The computational study was performed using a pressure-based solver in a fully coupled scheme. The implicit discretization was applied to the governing equations. For the gradient of the variables, the least squares cell-based method was selected. To obtain accurate results, the second-order upwind scheme was chosen. To maintain the stability of the simulation, the time step was determined by using the Courant-Friedrichs-Levy (CFL) number:

$$\mathit{CFL}=\frac{U_0\mathrm{\Delta }t}{\mathrm{\Delta }x}$$E5

where $U_0$ is the reference velocity on the wing, $\mathrm{\Delta }t$ is the time step and $\mathrm{\Delta }x$ is the average mesh cell size. Transient simulations were carried out for 5 cycles.

3.1 Validation of flapping wing kinematics

The validation of the numerical setup was performed using the data of a fruit fly experiment conducted by Dickinson et al. [1]. The kinematic pattern of the flapping wing was a sinusoidal motion for stroke position and angle of attack. A dynamically-scaled model of a fruit fly (Drosophila melanogaster) was built to imitate the kinematics of Drosophila wings and obtain aerodynamic forces over the wing using sensors. In the experiment, the motion of two wings was powered by computer-controlled motors and gearboxes attached to the wings. Furthermore, to reduce the wall effects, a 1 m $\times$ 1 m $\times$ 2 m tank was designed in which the wings were plunged in mineral oil with a density of 880 kg/m ${}^3$ and kinematic viscosity of 115 cSt.

The Drosophila wing used in the experiment was constructed from 25 cm long and 3.2 mm thick Plexiglas. The wing rotation axis was $0.2c$ from the leading edge, where $c$ is the chord length. The wing kinematics was described by using stroke amplitude, $\varphi ={160}^{\circ }$, angle of attack, $\alpha ={40}^{\circ }\pm 5^{\circ }$ at mid-stroke for both downstroke and upstroke, and wing-beat frequency, $f=145$ mHz.

In the numerical study, the domain size, wing morphology, fluid properties and wing kinematics were consistent with those in the experimental study. The wing, overset and computational setup are shown in Figure 3, where symmetry is assumed. Background ((Figure 3(a)) and overset (Figure 3(b)) domains include 770 k and 1.4 m polyhedral cells, respectively. During the wing movement, the stroke angle changed:

Figure 3.

(a) Computational setup with overset domain and symmetry walls, and (b) fruit fly wing planform used in the study with the meshed overset domain.

$$\varphi \left(t\right)=\frac{{\varphi }_{\mathit{max}}}{1.7424}\left[1-\left(2{cos}^{-1}\left(0.98sin\left(2\mathit{\pi ft}\right)\right)\right)\right]$$E6

where ${\varphi }_{\mathit{max}}={160}^{\circ }$ was the maximum stroke angle. The angle of attack of the wing varies:

$$\alpha \left(t\right)=\frac{{\alpha }_{\mathit{max}}}{tanh{C}_{\theta }}\left[-tanh\left(C_{\theta }cos\left(2\mathit{\pi ft}\right)\right)\right]$$E7

where ${\alpha }_{\mathit{max}}={40}^{\circ }$ at mid-stroke and $C_{\theta }=2.3$. The parameter $C_{\theta }$ determined the function deviation from sine to square waveform. The function transformed into a square wave, while $C_{\theta }$ approached infinity ($C_{\theta }\to \infty$). On the other hand, as $C_{\theta }$ approached zero ($C_{\theta }\to 0$), the function became a sine wave.

In the flapping wing, viscous stress and pressure over the wing surface govern the aerodynamic forces. In the global coordinate system (x, y, z), the lift force, $F_L$ is orthogonal to the stroke plane (x-z). The drag force, $F_D$ is aligned in the y-direction, and is parallel to the stroke plane [3]. Therefore, the lift and drag forces can be decomposed into the coordinate forces; $F_x$, $F_y$, $F_z$. The lift is related to:

$$F_L=F_y$$E8

and the drag is:

$$F_D=F_{x}cos\varphi -F_{z}sin\varphi$$E9

The lift and drag coefficients are respectively:

$$C_L=\frac{F_L}{0.5\rho U_0^{2}S}$$E10

$$C_D=\frac{F_D}{0.5\rho U_0^{2}S}$$E11

where $S$ is the wing area. The reference velocity is defined as, $U_0=2\mathit{\pi fR}$, where $R$ is the length of the wing.

The comparison of the lift and drag forces with the experimental data by Dickinson et al. [1] is shown in Figure 4, where $\tau =0-0.5$ is downstroke and $\tau =0.5-1$ is upstroke. The percent error obtained for the average lift and drag forces during a cycle are 3.4 and 7.2%, respectively.

Figure 4.

Comparison of drag and lift forces with the experimental data by Dickinson et al. [1].

3.2 Bumblebee kinematics

The present study utilizes the kinematic data of bumblebees in forward flight adopted from Xiong and Sun [36]. The kinematic data including amplitude, angle of attacks during the downstroke and upstroke, stroke plane deviation, the body angle and other parameters of bumblebee were calculated for different forward flights. Table 1 shows the kinematic parameters selected for 0 m/s (hovering), 1 m/s and 2.5 m/s. Based on the Dudley and Ellington [37], and Xiong and Sun [36], the duration of wing flip is 0.22 of the cycle time at stroke reversal.

U (m/s)	f (Hz)	$\varphi \left({}^{\circ }\right)$	$\overline{\varphi }\left({}^{\circ }\right)$	$\beta \left({}^{\circ }\right)$	${\alpha }_d\left({}^{\circ }\right)$	${\alpha }_u\left({}^{\circ }\right)$	$\chi \left({}^{\circ }\right)$	J
0	155	116	1	6	27	21	46.8	0
1	150	115	16	21.6	28.5	24	31.8	0.12
2.5	150	115	23	28	24.5	34	25	0.31

Table 1.

Kinematic parameters of bee for different forward velocities adopted from ref. [36].

During the wing motion, the stroke angle ($\varphi$) and angle of attack ($\alpha$) with respect to wing flip are defined by a sinusoidal function:

$$\varphi \left(t\right)=\overline{\varphi }+\varphi sin2\mathit{\pi ft}$$E12

$$\alpha \left(t\right)=\left\{\begin{array}{ll}{\alpha }_d+\frac{2{\alpha }_d}{\tau }\left(t-\frac{\tau }{2}-\frac{\tau }{2\pi }sin\left(2\pi \frac{t-0.5\tau }{\tau }\right)\right),& 0<t<\frac{\tau }{2}\\ {\alpha }_d,& \frac{\tau }{2}<t<t_1\\ {\alpha }_d+\frac{{\alpha }_u-{\alpha }_d}{\tau }\left(t-t_1-\frac{\tau }{2\pi }sin\left(2\pi \frac{t-t_2}{\tau }\right)\right),& t_1<t<t_2\\ {\alpha }_u,& t_2<t<t_3\\ {\alpha }_u-\frac{2{\alpha }_u}{\tau }\left(t-t_3-\frac{\tau }{2\pi }sin\left(2\pi \frac{t-t_3}{\tau }\right)\right),& t_3<t<1\end{array}\right\}$$E13

where, $t_1=0.5\left(1-\tau \right)$, $t_2=t_1+\tau$ and $t_3=1-0.5\tau$, with the Reynolds number of:

$$\mathit{Re}=\frac{\rho U_{0}R}{\mu }$$E14

where $R$ is the length of wing ($R=13.2$ mm) and $\mu$ is the dynamic viscosity of air. The maximum Reynolds number in the study is 7357.

3.3 Computational domain and overset mesh

The overset mesh, also named overlapping mesh, allows an analysis over multiple bodies that are moving or stationary. The method is an efficient way to both deal with the error related to moving boundaries and to circumvent the time-consuming complex mesh [38]. Moreover, it is easy to use, to sustain grid quality during the motion, and to avoid re-meshing malfunctions as well as generating simplified meshes for complex parts [39].

In the study, multiple overset mesh zones were used to simulate the wings and the body. The computational setup shown in Figure 5(a) has the dimensions of a $21R\times 17R\times 16R$ where the overset domain sizes of the wing and the body were $2.27R\times 1.81R\times 1.36R$ and $2.27R\times 1.51R\times 1.51R$, respectively (Figure 5(b)). The wing zone has a high-resolution grid than the body zone to capture the wake around the wing, as the background zone has coarser than the body zone.

Figure 5.

(a) Computational domain with the bee and (b) overset mesh showing how the bee and wings are carefully resolved.

This study considers two different models: thoracic and no thoracic models. In the thoracic model, the bee body is defined at a body temperature ($T_b$), with the bee transferring heat to the environment. The air properties change with ambient temperature ($T_a$) by specifying a polynomial relationship for the air dynamic viscosity. In the no thoracic model, the energy equation is not employed and therefore there is no heat transfer or dependence on temperature gradients. However, the fluid properties are specified using the ambient temperature for each case. The boundary conditions for the domain simulate a bee flying. The fluid velocity specified at the front of the domain (inlet) represents the motion of the bee. The other five sides of the domain represent ambient conditions and therefore zero gage pressure is specified (outlet).

3.4 Grid Independence test

A systematic procedure to test grid independence is a vital procedure to determine the optimal grid size, thus guaranteeing that the accuracy of the numerical solution is independent of mesh size. In this study, the grid convergence index (GCI) methodology by Celik et al. [40] was used.

Three mesh samples named ${\mathit{\text{coarse}}}_3$, ${\mathit{\text{medium}}}_2$ and ${\mathit{\text{fine}}}_1$ were prepared for the analysis. As shown in Table 2, $N$ represents the number of elements, and the grid refinement ratio $r$ is the ratio of the average cell size height of two different grids. The grid convergence index GCI is defined as:

Grid	$N$	$r$	${\overline{C}}_L$	GCI (%)
${\mathit{\text{Coarse}}}_3$	1.2 M	$r_{32}=1.312$	${\overline{C}}_{L,3}=1.2886$	${\mathit{GCI}}_{32}=0.867$
${\mathit{\text{Medium}}}_2$	2.8 M	$r_{21}=1.323$	${\overline{C}}_{L,2}=1.2708$	${\mathit{GCI}}_{21}=0.243$
${\mathit{\text{Fine}}}_1$	6.4 M		${\overline{C}}_{L,1}=1.2766$

Table 2.

GCI methodology parameters.

$$\mathit{GCI}=\frac{1.25}{r^p-1}\left|\frac{f_1-f_2}{f_1}\right|$$E15

where $p$ is the order of accuracy and $r$ is the grid refinement factor, defined as:

$$p=\frac{1}{lnr}ln\left|\frac{f_3-f_2}{f_2-f_1}\right|$$E16

$$r=h_2/h_1$$E17

The variable $h$ is the average mesh size and is calculated as:

$$h={\left[\frac{1}{N}\sum _{i=1}^N\left(\mathrm{\Delta }{V}_i\right)\right]}^{\frac{1}{3}}$$E18

where $\mathrm{\Delta }$$V_i$ is the volume of the i${}^{\mathit{th}}$ cell.

GCI analysis was performed by excluding the bee body; hence it only includes the total mesh count of the background and wing zone. To determine the GCI index, the average lift coefficient of a bumblebee hovering was considered. The ${\mathit{GCI}}_{32}=0.867\%$ for the coarse-medium comparison grids and ${\mathit{GCI}}_{21}=0.243\%$ for the medium-fine comparison (see Table 2). The lift coefficient of these three meshes for a cycle is shown in Figure 6 and the differences are minimal. Hence, the medium grid is chosen as the base mesh for subsequent simulations.

Figure 6.

Lift coefficient of the wing for different meshes at hovering position.

4. Results and discussions

The aerodynamic behavior of bumblebees for different environmental conditions was analyzed using a morphologically accurate bumblebee model and computational fluid dynamics. The effects of heat transfer between bee body and environment, and ambient conditions were analyzed for different temperature levels at three different forward speeds. The findings of the study are detailed with average lift-thrust forces, aerodynamic power, the time history of the forces and the power for a cycle and the visualization of temperature, pressure, and iso-surfaces with Q-criterion.

The flapping wing motion of the bee for a cycle is illustrated in Figure 7, where $\tau =0-0.5$ refers to downstroke and $\tau =0.5-1$ represents upstroke. During downstroke, the wing moves forward while it moves backward during upstroke in relation to the bee’s longitudinal axis. At the end of each stroke, the wing performs a rapid rotation and moves in the reverse direction.

Figure 7.

Kinematics of the flapping wing of the bee for a cycle, where $\tau =0-0.5$ is downstroke and $\tau =0.5-1$ is upstroke.

The transient simulations were performed for 5 cycles. Figure 8 shows the time history of lift ($F_L$) and thrust ($F_T$) forces of the thoracic model at 1 m/s and $T_b={32}^{\circ }C$ and $T_a={20}^{\circ }C$ for five cycles. The figure for the no thoracic model is virtually the same as the thoracic model, therefore only the thoracic model is shown in Figure 8. Even though there are no major differences in the cycles, in order to ensure transient effects are minimized, the third cycle ($\tau =0-1$) was chosen as the data used in the tables and figures for the subsequent simulations.

Figure 8.

Time history of lift and thrust forces at 1 m/s and $T_b={32}^{\circ }C$ and $T_a={20}^{\circ }C$ for five cycles.

The time history of lift and thrust forces of no thoracic model for a cycle is shown in Figure 9 at 0, 1, and 2.5 m/s for the different ambient temperatures ($T_a$). The downstroke is the most valuable part of lift production due to a larger leading-edge vortex (LEV) formation. At forward velocities, the majority of lift is produced during downstroke, but at hovering is produced during downstroke and upstroke. Bee flying at hovering produces less lift and thrust forces than forward speeds due to the absence of forward velocity.

Figure 9.

Time history of lift and thrust forces for a cycle at 0 m/s (hovering), 1 and 2.5 m/s for different ambient temperatures.

The thrust production mainly occurs during the upstroke and increases with increasing forward velocity. It is observed that during the downstroke, the negative thrust (drag) has a positive correlation, but it decreases during stroke reversal ($\tau =0.4-0.6$) with increasing forward flight. As shown in Figure 9, the increasing ambient temperature causes a drop in the lift and thrust at all speed levels. The majority of the lift loss takes place during the downstroke, where the main part of the lift is produced. In the upstroke, there is a decrease in the lift with increasing temperature, but it is not an apparent difference except in the hovering model. The thrust also drops with ambient temperature changes in the upstroke. However, the drag in the upstroke decreases with increasing temperature, which is higher at the mid-downstroke in the hover position.

The average lift and thrust forces are shown in Table 3 for thoracic and no thoracic models. The base cases assume negligible heat transfer, while two models at every velocity level are considered for the thoracic activity with ($T_b={32}^{\circ }C$) and ($T_b={38}^{\circ }C$) varying ambient temperatures ($T_a$). Increasing forward velocity increases the average lift and thrust forces in all conditions; thus, flying at 2.5 m/s has higher lift and thrust than 1 and 0 m/s. The bee experiences negative thrust (drag) force while hovering due to the absence of forward velocity. It is evident from the table that the average $F_L$ and $F_T$ for the models with $T_b={32}^{\circ }C$ and $T_a={20}^{\circ }C$, and $T_b={38}^{\circ }C$ and $T_a={28}^{\circ }C$ show minimal changes compared to the models without a thoracic region, thus indicating the limited impact of heat transfer due to the bee’s high frequency flapping to allow density changes.

	Temperature (${}^{\circ }C$)	0 m/s	1 m/s	2.5 m/s
${\overline{F}}_L$ ($\times {10}^{4}N$)	$T_a=10$	8.242	9.611	10.907
	$T_a=20$	7.945	9.260	10.491
	$T_a=20$, $T_b=32$	7.943	9.250	10.478
	$T_a=28$	7.718	9.001	10.194
	$T_a=28$, $T_b=38$	—	8.970	10.157
	$T_a=36$	7.499	8.692	9.826
${\overline{F}}_T$ ($\times {10}^{4}N$)	$T_a=10$	−0.937	0.0181	0.637
	$T_a=20$	−0.901	0.0165	0.601
	$T_a=20$, $T_b=32$	−0.903	0.0148	0.596
	$T_a=28$	−0.873	0.0162	0.581
	$T_a=28$, $T_b=38$	—	0.0153	0.573
	$T_a=36$	−0.846	0.0154	0.541

Table 3.

Average lift and thrust forces for different environmental conditions at 0, 1, and 2.5 m/s.

In contrast to thoracic activity, the effect of ambient temperature is prevalent. The higher ambient temperatures result in reduced lift and thrust forces. When $T_a$ increases from ${10}^{\circ }C$ to ${36}^{\circ }C$, the lift decreases by 9% for hovering, 9.6% for 1 m/s and 10% for 2.5 m/s, revealing that the bee experiences more lift loss flying at higher speeds. Moreover, the thrust in the hover position falls by 14.97 and 15% for 1 and for 2.5 m/s, respectively. Contrary to forward flight, the bee experiences negative thrust (drag) in hovering due to the lack of forward velocity. The drag during hovering decreases by 9.71% when the ambient temperature increases from ${10}^{\circ }C$ to ${36}^{\circ }C$. As a result, warmer environments negatively affect the bee overall performance, while the effect of the thoracic activity is limited.

The temperature contours of the thoracic model ($T_b={32}^{\circ }C,T_a={20}^{\circ }C$) are shown in Figure 10 at 2.5 m/s. Figure 10(a) shows temperature changes around the bee. The heat plume from the body alters the flow conditions by increasing the air temperature around the bee, resulting in density reduction and viscosity increase of air. The distribution of temperature over the wing dorsal surface is shown in Figure 10(b). The heat plume emitted from the body increases the temperature on the forewing and hindwing. Specifically, the temperature on the leading-edge and upper part of the hindwing regions are similar, as the temperature increase on the trailing-edge of the hindwing is higher. As mentioned above, due to the limited effect of the thoracic activity, the remainder of the discussion and results are for the “no thoracic model” in which only ambient temperature is changed.

Figure 10.

Temperature contours showing the flow conditions around the bee (a) and the temperature distribution on the wing dorsal side (b) at 2.5 m/s ($\tau =0.33$) for $T_b={32}^{\circ }C,T_a={20}^{\circ }C$.

Figure 11 depicts the suction pressure (negative pressure because it is gage pressure) along the dorsal side of the wing during the downstroke at a time instant of $\tau =0.25$ for the lowest (${10}^{\circ }C$) and highest (${36}^{\circ }C$) ambient temperatures at 0, 1, and 2.5 m/s. The bee at 2.5 m/s has lower suction pressure on the dorsal due to high spanwise velocity than 0 m/s and 1 m/s, leading to larger lift production. The results show that there are changes in the pressure distribution on the forewing and hindwing as the ambient temperature increases from ${10}^{\circ }C$ to ${36}^{\circ }C$, resulting in increasing suction pressure at all speed levels, resulting in decreasing the lift. Particularly, the low-pressure regions on the leading-edge, wingtip, and hindwing are reduced, resulting in a decrease in the lift force. Furthermore, there are pressure increases spotted on the root side of the wing for 1 and 2.5 m/s, which also contribute to lift reduction.

Figure 11.

Pressure (gage) contours on the dorsal side of the wing for different temperature and velocity levels at mid-downstroke ($\tau =0.25$).

Additionally, the pressure distribution on the ventral side during the upstroke at $\tau =0.75$ for the lowest and the highest ambient temperatures is depicted in Figure 12. The figure only shows the extreme cases of 0 and 2.5 m/s. For hovering, lower pressure is experienced on the wing tip of the ventral, leading to more lift during the upstroke, proving that more lift production takes place at hovering during upstroke and downstroke than forward flight. When the ambient temperature increases from ${10}^{\circ }C$ to ${36}^{\circ }C$, the differences in pressure are greater at 2.5 m/s compared to 0 m/s. The reduction in the size of the negative pressure zone in the wingtip area, as a result of increased ambient temperature, is associated with decreased lift and thrust, while there are no significant changes in the rest of the wing surface.

Figure 12.

Pressure (gage) contours on the ventral side of the wing for different temperature and velocity levels at 0 and 2.5 m/s at mid-upstroke ($\tau =0.75$).

Figure 13 displays the vortex structures with pressure contour following stroke reversal using the Q-criterion at $\tau =0.33$ for 1 m/s. LEV formation during the downstroke makes an important contribution to the lift by creating suction pressure on the dorsal side of the wing. Increasing ambient temperature does not change LEV but small changes in the trailing edge vortex (TEV) structures. Furthermore, the suction pressure on the leading-edge and the hindwing increases with increasing ambient temperature, thereby leading to reducing lift.

Figure 13.

Iso-surfaces with Q-criterion and pressure distribution ($5\times {10}^6$) at 1 m/s ($\tau =0.33$).

The iso-surfaces with pressure (gage) contours for the lowest and the highest temperatures at 0 and 2.5 m/s are presented in Figure 14. At higher velocities, a thicker tip vortex (TV) is generated due to the availability of strong spanwise flow, so the 2.5 m/s model predicts lower pressure over the wing than 0 m/s. The trailing-edge vortex (TEV) structures at 0 m/s are larger than at 2.5 m/s, leading to higher lift and drag. When ambient temperature increases from ${10}^{\circ }C$ to ${36}^{\circ }C$, vortex structures do not change considerably for 0 and 2.5 m/s. On the wing leading-edge and hindwing, while there is no significant increase in pressure at 0 m/s, the suction pressure at 2.5 m/s is considerably affected by increasing ambient temperature, resulting in more lift drop. The Q-criterion iso-surfaces with vorticity magnitude for the lowest and highest temperatures during upstroke are shown in Figure 15 at 2.5 m/s. Small changes are observed in iso-surfaces and vorticity magnitude with increasing temperature that result in slight differences in lift and thrust at $\tau =0.65$.

Figure 14.

Iso-surfaces with Q-criterion and pressure distribution ($3\times {10}^6$) at 0 and 2.5 m/s ($\tau =0.33$).

Figure 15.

Iso-surfaces with Q-criterion and vorticity magnitude ($3\times {10}^6$) at 1 m/s ($\tau =0.65$).

Figure 16 illustrates the iso-surfaces with vorticity magnitude at 2.5 m/s and different time periods ($\tau$) for the lowest and highest ambient temperatures. Note that there are no major changes in the vorticity structures and magnitudes except at $\tau =0.55$, in which iso-surfaces at the wingtip disappeared when the ambient temperature increased from ${10}^{\circ }C$ to ${36}^{\circ }C$.

Figure 16.

Iso-surfaces with Q-criterion and vorticity magnitude ($4\times {10}^6$) at 2.5 m/s for different time periods in a cycle.

Bees alter their wing beat frequency when they need to increase or decrease lift and thrust forces. Under the same kinematic parameters, a bee flying in warmer environments must change its flapping frequency to maintain flight performance. When ambient temperature increases from ${10}^{\circ }C$ to ${36}^{\circ }C$, the bee must increase its flapping frequency to compensate for the lift loss. The average lift forces of the bee at the lowest and highest temperatures for three different velocities are shown in Table 4 with the adjusted frequencies ($f_{\mathit{adj}}$) used in the simulations. Using the lift forces, an interpolation was performed to determine the adjusted frequencies of the bee flying at ${36}^{\circ }C$. Then these frequencies were validated by performing the new simulations, which proved that the lift lost at ${36}^{\circ }C$ was regained using the adjusted frequencies. The difference in the lift between the original and adjusted frequency is 0.4% for 0 m/s, 0.05% for 1 m/s and 0.2% for 2.5 m/s. To maintain the lift requirement flying at ${36}^{\circ }C$ ambient, the bee must increase the flapping frequency from 155 to 162 Hz for 0 m/s, 150 to 158 Hz for 1 m/s, and 150 to 159 Hz for 2.5 m/s. The difference in the flapping frequency for ${10}^{\circ }C$ and ${36}^{\circ }C$ is 4.5% for 0 m/s, 6% for 1 m/s, and 5.3% for 2.5 m/s.

	Temperature	0 m/s	1 m/s	2.5 m/s
${\overline{F}}_L$ ($\times {10}^4$ N)	$T_a={10}^{\circ }C$	8.242	9.611	10.907
		($f=155$ Hz)	($f=150$ Hz)	($f=150$ Hz)
	$T_a={36}^{\circ }C$	7.499	8.692	9.826
		($f=155$ Hz)	($f=150$ Hz)	($f=150$ Hz)
	$T_a={36}^{\circ }C$	8.208	9.606	10.937
		($f_{\mathit{adj}}=162$ Hz)	($f_{\mathit{adj}}=159$ Hz)	($f_{\mathit{adj}}=158$ Hz)

Table 4.

Average lift forces with the adjusted frequency of the lowest (${10}^{\circ }C$) and the highest ambient temperature (${36}^{\circ }C$) models at 0, 1, and 2.5 m/s.

Aerodynamic power for flapping flight represents the power to overcome the fluid forces, defined as $P=\mathit{M\omega }$, where $M$ and $\omega$ are the moment and the angular velocity of the wing. The time history of the aerodynamic power of the bee at different forward speeds is shown in Figure 17 for the lowest and highest ambient temperature with the original and adjusted frequencies. The majority of power production takes place during the downstroke and increasing forward speed increases the aerodynamic power; therefore, the 2.5 m/s model generates more power during the downstroke. Moreover, the power reduction during stroke reversal ($\tau =0.4-0.6$) is maximum at 2.5 m/s and minimum at 1 m/s, indicating a higher average power generation of 1 m/s than 0 m/s and 2.5 m/s. In the upstroke, the power generation at 0 m/s is higher than at 1 m/s but lower than at 2.5 m/s. A larger decline in power at 0 m/s was also observed at the end of the upstroke. Increasing temperature decreases the power production during downstroke and upstroke at all forward speeds but is maximum at mid-downstroke. The adjusted wing-beat frequencies shown in Table 4 compensate for the power at all speed levels that was lost with the ambient temperature rise from ${10}^{\circ }C$ to ${36}^{\circ }C$. The power increase is mostly observed during the downstroke and upstroke, and it is maximum at mid-downstroke.

Figure 17.

Time history of aerodynamic power of ${10}^{\circ }C$ and ${36}^{\circ }C$ models for one cycle at 0, 1, and 2.5 m/s for the original and adjusted frequencies.

The average aerodynamic power of the lowest and highest temperature models with the original and adjusted frequency model is illustrated in Table 5. The average aerodynamic power of the bee at 1 m/s is higher than at 0 and 2.5 m/s. When ambient temperature increases from ${10}^{\circ }C$ to ${36}^{\circ }C$, the aerodynamic power drops by 7.9% for 0 m/s, 8% for 1 m/s and 7.3% for 2.5 m/s, indicating that the bee suffers more aerodynamic power loss at low speeds. The bee employs the adjusted frequencies to compensate for the loss. The difference in the average aerodynamic power between ${10}^{\circ }C$ model and the adjusted frequency model of ${36}^{\circ }C$ is 5% for 0 m/s, 6% for 1 m/s, and 12% for 2.5 m/s. Thus, the adjusted frequencies require more power for the 2.5 m/s model.

	Temperature	0 m/s	1 m/s	2.5 m/s
$\overline{P}$ (mW)	$T_a={10}^{\circ }C$ ($f_{\mathit{\text{orig}}.}$)	3.8	5.0	4.1
	$T_a={36}^{\circ }C$ ($f_{\mathit{\text{orig}}.}$)	3.5	4.6	3.8
	$T_a={36}^{\circ }C$ ($f_{\mathit{adj}.}$)	4.0	5.3	4.6

Table 5.

Average aerodynamic power of ${10}^{\circ }C$ and ${36}^{\circ }C$ models at 0, 1, and 2.5 m/s for the original and adjusted frequencies.

5. Conclusion

This study investigated the impacts of heat transfer and ambient temperature changes on bee flight performance, highlighting a better understanding of the roles of thoracic and ambient temperature fluctuations on bumblebee aerodynamic performance. Heat transfer resulting from thoracic temperature activity has a rather limited impact on bee performance as could be detected by the rigid-wing models. It was observed that heat transfer with the environment increases the temperature of the wing surface and changes the flow conditions around the bee. Hence, the lift and thrust forces of the thoracic model decreased when compared to the no thoracic model for the same ambient condition, indicating that thoracic activity has an impact, however limited.

Ambient temperature change has more impact on the aerodynamics of the bee than thoracic temperature activity. Increasing ambient temperature changes the flow properties, thereby resulting in lower lift. Thrust force reduces at 1 m/s and 2.5 m/s, as the drag (negative thrust at hovering) becomes lower at 0 m/s (hovering) with increasing ambient temperature. Increasing temperature from the lowest temperature (${10}^{\circ }C$) to the highest temperature (${36}^{\circ }C$) caused 9, 9.6, and 10% less lift at 0, 1, and 2.5 m/s, respectively, underlining more lift loss at higher velocities. The thrust dropped 14.97 and 15% at 1 and 2.5 m/s; on the contrary, the drag decreased at hovering by 9.71%. Furthermore, the aerodynamic power decreased by 7.9% for 0 m/s, 8% for 1 m/s and 7.3% for 2.5 m/s when ambient temperature increased from ${10}^{\circ }C$ to ${36}^{\circ }C$. To compensate for the lift loss, the adjusted frequency was specified. The lift losses could be recovered, while increasing the aerodynamic power by 5% for 0 m/s, 6% for 1 m/s and 12% for 2.5 m/s.

As a result, warmer temperatures reduce bee performance by negatively affecting thoracic temperature and flow condition variations around the bees. Heat transfer resulting from thoracic and environmental temperature changes, reduces the aerodynamic performance of bees. While the environmental effects in bee performance can be clearly determined, the real influence of thoracic temperature change cannot be accurately analyzed without improving the computational model. For example, laboratory experiments that can measure the flow physics when a bee is subjected to different environmental temperatures can provide data to verify the models and guide model improvements. Overall, it can be deduced that bees are able to compensate loss of lift and power due to increasing temperatures with a rise in their flapping frequency. However, it is envisaged that rising environmental heat, requiring higher wing beats, will result in increased thoracic temperature. There comes a time that the combined temperature effects supersede the bee’s muscular abilities to compensate for lift and power, hence, drastically reducing the bee’s endurance or limiting its foraging perimeter.