The penetration of a spherical vortex into turbulence is studied theoretically and experimentally. The characteristics of the vortex are first analysed from an integral perspective that reconciles the far-field dipolar flow with the near-field source flow. The influence of entrainment on the vortex drag force is elucidated, extending the Maxworthy (J. Fluid Mech., vol. 81, 1977, pp. 465–495) model to account for turbulent entrainment into the vortex movement and vortex penetration into an evolving turbulent field. The physics are explored numerically using a spherical vortex (initial radius $R_0$, speed $U_{v0}$), characterised by a Reynolds number $Re_0(=2R_0U_{v0}/\nu$, where $\nu$ is the kinematic viscosity) of 2000, moving into decaying homogeneous turbulence (root-mean-square $u_0$, integral scale $L$) with turbulent intensity $I_t=u_0/U_{v0}$. When the turbulence is absent ($I_t=0$), a wake volume flux leads to a reduction of vortex impulse that causes the vortex to slow down. In the presence of turbulence ($I_t> 0$), the loss of vortical material is enhanced and the vortex speed decreases until it is comparable to the local turbulent intensity and quickly fragments, penetrating a distance that scales as $I_t^{-1}$. In the experimental study, a vortex ($Re_0\sim 1490\unicode{x2013}5660$) propagating into a statistically steady, spatially varying turbulent field ($I_{ve}=0.02$ to 0.98). The penetration distance is observed to scale with the inverse of the turbulent intensity. Incorporating the spatially and temporally varying turbulent fields into the integral model gives a good agreement with the predicted trend of the vortex penetration distance with turbulent intensity and insight into its dependence on the structure of the turbulence.

Turbulent flow around curved tandem cylinders has been studied for the first time, by means of direct numerical simulation. The convex configuration was used, with a nominal gap ratio of $L/D = 3$ and a Reynolds number of 3900. Along the span, the flow regimes vary from alternating overshoot/reattachment to co-shedding. Three distinct Strouhal numbers coexist in the flow that are tied directly to different tandem cylinder flow regimes. This result differs substantially from convex curved tandem cylinders at a transitional Reynolds number, where only a single dominant frequency is found. All regimes exhibit some degree of instability, so that the flow can be considered multistable. A mode switch from alternating overshoot/reattachment to symmetric reattachment is found. Complex interactions are observed between the primary instability, the shear layer instability and the flow mode alterations. As opposed to previous investigations with single and tandem straight cylinders in the subcritical flow regime, our results indicate that there may be direct feedback from the primary instability to the shear layer instability. The downdraft region in the gap exhibits slow meandering, and may travel upstream and amplify the shear layer instability, causing early transition in the gap shear layer. This downdraft is governed by the slow modulations of the vortex formation region in the lower gap, meaning that the vortex dynamics of this region may indirectly influence the shear layer instability higher up in the gap.

This research investigates the wake–foil interactions between two oscillating foils in a tandem configuration undergoing energy harvesting kinematics. Oscillating foils have been shown to extract hydrokinetic energy from free-stream flows through a combination of periodic heave and pitch motions, at relatively higher amplitudes and lower reduced frequency than thrust generating foils. When placed in tandem, the wake–foil interactions can govern the energy harvesting efficiency of the system due to a reduced relative flow velocity in combination with a structured and coherent wake of vortices shed from the high amplitude flapping of upstream foils. This work utilizes simulations of two tandem foils to parameterize and model the energy harvesting performance as a function of array configuration and foil kinematics. Once the wake of the leading foil has been fully parameterized, the placement, phase angle and kinematic stroke of the second foil is utilized to estimate the time-dependent power curve. The algorithm predicts the power of the second foil through the mean and unsteady wake characteristics, including the direct impingement of a vortex with the trailing foil.

Birds employ rapid pitch-up motions close to the ground for different purposes: perching birds use this motion to decelerate and come to a complete stop while hunting birds, such as bald eagles, employ it to catch prey and swiftly fly away. Motivated by these observations, our study investigates how natural flyers accomplish diverse flying objectives by rapidly pitching their wings while decelerating near ground. We conducted experimental and analytical investigations focusing on rapidly pitching plates during deceleration in close proximity to the ground to explore the impact of ground proximity on the unsteady dynamics. Initially, we executed synchronous pitch-up motion, where both pitching and deceleration have the same motion duration, at different ground heights. Experimental results demonstrate that as the pitching wing approaches the ground, the instantaneous lift increases by approximately $38\,\%$ compared with a far-from-ground case, while the initial peak drag force remains relatively unchanged. Our analytical model conforms to this trend, predicting an increase in lift force as the wing approaches the ground, indicating enhanced added mass and circulatory lift force due to the ground effect. Next, we examined asynchronous pitch-up motion cases, where rapid pitching motions were initiated at different stages of deceleration. The results reveal that initiating the wing pitch early in the deceleration leads to the formation of larger counter-rotating vortices at the early stage of the manoeuvre. These vortices generate stronger dipole jets that orient backward in the later stages of the manoeuvre after impinging with the ground surface, which hunting birds utilize to accelerate after catching prey. Conversely, when the wing pitch is delayed, smaller vortices form, but their growth is postponed until late in the manoeuvre. This delayed vortex growth produces lift and drag force at the end phase of the manoeuvre that facilitates a smooth landing or perching. Thus, through strategic tuning of a rapid pitch-up motion with deceleration, natural flyers, such as birds, achieve diverse flying objectives.

We present a systematic simulation campaign to investigate the pairwise interaction of two mobile, monodisperse particles submerged in a viscous fluid and subjected to monochromatic oscillating flows. To this end, we employ the immersed boundary method to geometrically resolve the flow around the two particles in a non-inertial reference frame. We neglect gravity to focus on fluid–particle interactions associated with particle inertia and consider particles of three different density ratios aligned along the axis of oscillation. We systematically vary the initial particle distance and the frequency based on which the particles show either attractive or repulsive behaviour by approaching or moving away from each other, respectively. This behaviour is consistently confirmed for the three density ratios investigated, although particle inertia dictates the overall magnitude of the particle dynamics. Based on this, threshold conditions for the transition from attraction to repulsion are introduced that obey the same power law for all density ratios investigated. We furthermore analyse the flow patterns by suitable averaging and decomposition of the flow fields and find competing effects of the vorticity induced by the fluid–particle interactions. Based on these flow patterns, we derive a circulation-based criterion that provides a quantitative measure to categorize the different cases. It is shown that such a criterion provides a consistent measure to distinguish the attractive and repulsive arrangements.

To optimize flapping foil performance, in the current study we apply deep reinforcement learning (DRL) to plan foil non-parametric motion, as the traditional control techniques and simplified motions cannot fully model nonlinear, unsteady and high-dimensional foil–vortex interactions. Therefore, a DRL training framework is proposed based on the proximal policy optimization algorithm and the transformer architecture, where the policy is initialized from the sinusoidal expert display. We first demonstrate the effectiveness of the proposed DRL-training framework, learning the coherent foil flapping motion to generate thrust. Furthermore, by adjusting reward functions and action thresholds, DRL-optimized foil trajectories can gain significant enhancement in both thrust and efficiency compared with the sinusoidal motion. Last, through visualization of wake morphology and instantaneous pressure distributions, it is found that DRL-optimized foil can adaptively adjust the phases between motion and shedding vortices to improve hydrodynamic performance. Our results give a hint of how to solve complex fluid manipulation problems using the DRL method.

We simulate the head-on collision between vortex rings with circulation Reynolds numbers of 4000 using an adaptive, multiresolution solver based on the lattice Green's function. The simulation fidelity is established with integral metrics representing symmetries and discretization errors. Using the velocity gradient tensor and structural features of local streamlines, we characterize the evolution of the flow with a particular focus on its transition and turbulent decay. Transition is excited by the development of the elliptic instability, which grows during the mutual interaction of the rings as they expand radially at the collision plane. The development of antiparallel secondary vortex filaments along the circumference mediates the proliferation of small-scale turbulence. During turbulent decay, the partitioning of the velocity gradients approaches an equilibrium that is dominated by shearing and agrees well with previous results for forced isotropic turbulence. We also introduce new phase spaces for the velocity gradients that reflect the interplay between shearing and rigid rotation and highlight geometric features of local streamlines. In conjunction with our other analyses, these phase spaces suggest that, while the elliptic instability is the predominant mechanism driving the initial transition, its interplay with other mechanisms, e.g. the Crow instability, becomes more important during turbulent decay. Our analysis also suggests that the geometry-based phase space may be promising for identifying the effects of the elliptic instability and other mechanisms using the structure of local streamlines. Moving forward, characterizing the organization of these mechanisms within vortices and universal features of velocity gradients may aid in modelling turbulent flows.

Over the last decade, substantial progress has been made in understanding the topology of quasi-two-dimensional (2-D) non-equilibrium fluid flows driven by ATP-powered microtubules and microorganisms. By contrast, the topology of three-dimensional (3-D) active fluid flows still poses interesting open questions. Here, we study the topology of a spherically confined active flow using 3-D direct numerical simulations of generalized Navier–Stokes (GNS) equations at the scale of typical microfluidic experiments. Consistent with earlier results for unbounded periodic domains, our simulations confirm the formation of Beltrami-like bulk flows with spontaneously broken chiral symmetry in this model. Furthermore, by leveraging fast methods to compute linking numbers, we explicitly connect this chiral symmetry breaking to the entanglement statistics of vortex lines. We observe that the mean of linking number distribution converges to the global helicity, consistent with the asymptotic result by Arnold [In Vladimir I. Arnold – Collected Works (ed. A.B. Givental, B.A. Khesin, A.N. Varchenko, V.A. Vassiliev & O.Y. Viro), pp. 357–375. Springer]. Additionally, we characterize the rate of convergence of this measure with respect to the number and length of observed vortex lines, and examine higher moments of the distribution. We find that the full distribution is well described by a k-Gamma distribution, in agreement with an entropic argument. Beyond active suspensions, the tools for the topological characterization of 3-D vector fields developed here are applicable to any solenoidal field whose curl is tangent to or cancels at the boundaries of a simply connected domain.

A numerical investigation was conducted on $Re_{\varGamma _{0}}=3000$ vortex rings colliding with wall-mounted hemispheres to study how their relative sizes affect the resulting vortex dynamics and structures. The hemisphere to vortex ring diameter ratio ranges from $D/d=0.5$ to $D/d=2$. Secondary/tertiary vortex rings are observed to result from hemispheric surface boundary layer separations rather than wall boundary layer separations as the diameter ratio increases. While those for $D/d\leq 1$ hemispheres can be attributed to sequential hemispheric and wall boundary layer separations, the primary vortex ring produces a series of secondary/tertiary vortex rings only along the $D/d=2$ hemispheric surface. This indicates that the presence of the wall makes little difference when the hemisphere is sufficiently large. On top of comparing vortex ring circulations and translational velocities between hemisphere and flat-wall based collisions, present collision outcomes have also been compared with those predicted by specific discharge velocity models. Additionally, comparisons of vortex core trajectories and vortex ring formation locations with earlier cylindrical convex surface based collisions provide more clarity on differences between two- and three-dimensional convex surfaces. Finally, vortex flow models are presented to account for the significantly different flow behaviour as the hemisphere size varies. Specifically, the vortex flow model for the $D/d=2$ hemisphere hypothesizes that the recurring tertiary vortex ring formations cease only when the primary vortex ring slows down sufficiently for the last tertiary vortex ring to entangle with it and render it incoherent. Until that happens, the primary vortex ring will continue to induce more tertiary vortex rings to form, with potential implications for heat/mass transfer optimizations.

The growth of perturbations subject to the Crow instability along two vortex rings of equal and opposite circulation undergoing a head-on collision is examined. Unlike the planar case for semi-infinite line vortices, the zero-order geometry of the flow (i.e. the ring radius, core thickness and separation distance) and by extension the growth rates of perturbations vary in time. The governing equations are therefore temporally integrated to characterize the perturbation spectrum. The analysis, which considers the effects of ring curvature and the distribution of vorticity within the vortex cores, explains several key flow features observed in experiments. First, the zero-order motion of the rings is accurately reproduced. Next, the predicted emergent wavenumber, which sets the number of secondary vortex structures emerging after the cores come into contact, agrees with experiments, including the observed increase in the number of secondary structures with increasing Reynolds number. Finally, the analysis predicts an abrupt transition at a critical Reynolds number to a regime dominated by a higher-frequency, faster-growing instability mode that may be consistent with the experimentally observed rapid generation of a turbulent puff following the collision of rings at high Reynolds numbers.

We present a systematic study to investigate the fluid–structure interaction (FSI) of subaqueous spherical pendulums with several solid-to-fluid mass ratios $m^*\in [1.14, 14.95]$ and corresponding Reynolds numbers of up to $\textit {Re}\sim 10^4$. A digital object tracking (DOT) method was employed to track the oscillating pendulum spheres whereas the time-resolved 3-D particle tracking velocimetry (tr-3D-PTV) was used to measure the flow field around the spheres. The data obtained from the coupling of the two measuring techniques provide novel insights into the dynamics of pendulum sphere oscillations, instantaneous pressure fluctuations related to vortex shedding around the spheres and the way they are influenced by the vortex and wake interactions. Namely, we show that during the downward motion of the pendulum spheres, vortex rings are shed off the spheres which, in turn, induce short-lived propulsion and, subsequently, distinct deceleration. Further, we used the measured data to improve an existing basic model of pendulum motion, which has significant discrepancies for the period and peak amplitude predictions. We did this by incorporating a vortex-induced drag term and a wake interaction term into the equation. Finally, the improved equations are shown to be capable of predicting the subaqueous pendulum dynamics with high accuracy, for the investigated range of $m^*$. The study thus extends the current understanding of basic fluid dynamic mechanisms such as added mass, nonlinear drag, vortex and pressure dynamics.

A transcritical domain with a sharp two-phase interface may exist during the early times of liquid hydrocarbon fuel injection at supercritical pressure. Thus, two-phase dynamics are sustained before substantial heating of the liquid and drive the early three-dimensional deformation and atomisation. A recent study of a transcritical liquid jet showed distinct deformation features caused by interface thermodynamics, low surface tension and intraphase diffusive mixing. In the present work, the compressible vortex identification method $\lambda _\rho$ is used to study the vortex dynamics in a cool liquid n-decane transcritical jet surrounded by a hotter oxygen gaseous stream at supercritical pressures. The relationship between vortical structures and the liquid surface evolution is detailed, along with the vorticity generation mechanisms, including variable-density effects. The roles of hairpin and roller vortices in the early deformation of lobes, the layering and tearing of liquid sheets and the formation of fuel-rich gaseous blobs are analysed. At these high pressures, enhanced intraphase mixing and ambient gas dissolution affect the local liquid structures (i.e. lobes). Thus, liquid breakup differs from classical sub-critical atomisation. Near the interface, liquid density and viscosity drop by up to 10 % and 70 %, respectively, and the liquid is more easily affected by the vortical motion (e.g. liquid sheets wrap around vortices). Despite the variable density, compressible vorticity generation terms are smaller than the vortex stretching and tilting. Layering traps and aligns the vortices along the streamwise direction while mitigating the generation of new rollers.

Flow around curved tandem cylinders in the convex configuration has been studied by means of direct numerical simulations, for a Reynolds number of 500 and a nominal gap ratio of 3.0. Spanwise variation of flow regimes, as well as curvature-induced axial velocity, leads to an exceedingly complex vortex dynamics in the wake. Both parallel and oblique vortex shedding are observed. Oblique shedding is connected to repeated occurrences of dislocations. The dislocations are caused by two main mechanisms: frequency differences in the upper part of the curved geometry and shedding of gap vortices into the lower near wake. Both types of dislocations are closely associated with a mode switch in the gap. In parts of the gap, there is low-frequency quasi-periodic asymmetry of the gap vortices, where the flow is biased to one side of the gap for intervals of several wake vortex shedding periods. The switch from side to side is associated with a surge of the vertical velocity, and the frequency of the switch is similar to that of long-term variation of the recirculation length in the lower gap.

The three-dimensional (3-D) transition of the leading-edge vortex (LEV) and the force characteristics of the plunging airfoil are investigated in the chord-based Strouhal number $St_c$ range of 0.10 to 1.0 by means of experimental measurements, numerical simulations and linear stability analysis in order to understand the spanwise instabilities and the effects on the force. We find that the interaction pattern of the LEV, the LEV from a previous cycle (pLEV) and the trailing-edge vortex (TEV) is the primary mechanism that affects the 3-D transition and associated force characteristics. For $St_c \leq 0.16$, the 3-D transition is dominated by the LEV–TEV interaction. For $0.16 < St_c \leq 0.44$, the TEV lies in the middle of the LEV and the pLEV and therefore vortex interaction between them is relatively weak; as a result, the LEV remains two-dimensional up to a relatively high Reynolds number of $Re = 4000$ at $St_c = 0.32$. For $0.44 < St_{c} \leq 0.54$, and at relatively low Reynolds numbers, the pLEV and the TEV tend to form a clockwise vortex pair, which is beneficial for the high lift and stability of the LEV. For $0.49 \leq St_c$, the pLEV and TEV tend to form an anticlockwise vortex pair, which is detrimental to the lift and flow stability. In the last $St_c$ range, vortex interaction involving the LEV, the TEV and the pLEV results in an unstable period-doubling mode which has a wavelength of about two chord-lengths and the 3-D transition enhances the lift.

The vortex dynamics and the structural load in a step cylinder (consisting of a small, d, and a large, D, cylinder) flow are investigated numerically at Reynolds number ($Re_D$) 150 for diameter ratios $D/d=2.0, 2.4$ and 2.8. First, the formation mechanism of a non-uniform oblique vortex shedding (the vortex shedding frequency remains unchanged as the oblique shedding angle varies) behind the small cylinder is explained: an increase in the production rate of the vortex strength and a farther downstream movement of the vortex formation position occur simultaneously as the vicinity of the step is approached along the small cylinder. Second, the structural load (the drag and lift) along the step cylinder is investigated, where four local extremes (two local minima and two local maxima) are observed. An in-depth investigation of the vortex dislocation effects on the structural load is provided, showing that the decreased circulation in the near wake and the weakened staggered Kármán vortex shedding pattern cause a major reduction (90 %) of the sectional lift amplitude and a relatively modest reduction (5.7 %) of the sectional drag amplitude, compared with the corresponding sectional force when no vortex dislocation occurs. This new knowledge combined with the three-dimensional effect of the step cylinder wake (caused by the blending of the small and larger cylinder wakes around the step) explain the formation of the four local extremes and the distribution of the structural load between them. Finally, it is found that the increasing $D/d$ amplifies the structural load variation along the step cylinder.

We investigate the nonlinear evolution of pairs of three-dimensional, equal-sized and opposite-signed vortices at finite Froude and Rossby numbers. The two vortices may be offset in the vertical direction. The initial conditions stem from relative equilibria obtained numerically in the quasi-geostrophic regime, for vanishing Froude and Rossby numbers. We first address the linear stability of the quasi-geostrophic opposite-signed pairs of vortices, and show that for all vertical offsets, the vortices are sensitive to an instability when close enough together. In the nonlinear regime, the instability may lead to the partial destruction of the vortices. We then address the nonlinear interaction of the vortices for various values of the Rossby number. We show that as the Rossby number increases, destructive interactions, where the vortices break into pieces, may occur for a larger separation between the vortices, compared to the quasi-geostrophic case. We also show that for well-separated vortices, the interaction is non-destructive, and ageostrophic effects lead to the deviation of the trajectory of the pair of vortices, as the anticyclonic vortex dominates the interaction. Finally, we show that the flow remains remarkably close to a balanced state, emitting only waves containing negligible energy, even when the interaction leads to the destruction of the vortices.

This study examines the two-dimensional interaction of two unequal co-rotating viscous vortices in uniform background shear. Numerical simulations are performed for vortex pairs having various circulation ratios $\varLambda _0 = \varGamma _{1,0}/\varGamma _{2,0} = (\omega _{1,0}/\omega _{2,0})(a^2_{1,0}/a^2_{2,0}) \leqslant 1$, corresponding to different initial characteristic radii $a_{i,0}$ and peak vorticities $\omega _{i,0}$ of each vortex $i=1,2$, in shears of various strengths $\zeta _0 = \omega _S/\omega _{2,0}$, where $\omega _S$ is the constant vorticity of the shear. Two primary flow regimes are observed: separations ($\zeta _0 < \zeta _{sep} < 0$), in which the vortices move apart continuously, and henditions ($\zeta _0 > \zeta _{sep}$), in which the interaction results in a single vortex (where $\zeta _{sep}$ is the adverse shear strength beyond which separation occurs). Vortex motion and values of $\zeta _{sep}(\varLambda _0)$ are well-predicted by a point-vortex model for unequal vortices. In vortex-dominated henditions, shear varies the peak–peak distance $b$, and vortex deformation. The main convective interaction begins when core detrainment of one vortex is established, and proceeds similarly to the no-shear ($\zeta _0 = 0$) case: merger occurs if the second vortex also detrains, engendering mutual entrainment; otherwise straining out occurs. Detrainment requires persistence of straining of both sufficient magnitude, as indicated by relative straining above a consistent critical value, $(S/\omega )_i > (S/\omega )_{cr}$, where $S$ is the strain rate magnitude at the vorticity peak, and conducive direction. Hendition outcomes are assessed in terms of an enhancement factor $\varepsilon \equiv \varGamma _{end}/\varGamma _{2,start}$. Although $\varepsilon$ generally varies with $\zeta _0$, $(a^2_{1,0} /a^2_{2,0} )$ and $(\omega _{1,0}/\omega _{2,0})$ in a complicated manner, this variation is well-characterized by the pair's starting enstrophy ratio, $Z_2/Z_1$. Within a transition region between merger and straining out (approximately $1.65 < Z_2/Z_1 < 1.9$), shear of either sense may increase $\varepsilon$.

New exact solutions are presented to the problem of steadily travelling water waves with vorticity wherein a submerged von Kármán point vortex street cotravels with a wave on a linear shear current. Surface tension and gravity are ignored. The work generalizes an earlier study by Crowdy & Nelson (Phys. Fluids, vol. 22, 2010, 096601) who found analytical solutions for a single point vortex row cotravelling with a water wave in a linear shear current. The main theoretical tool is the Schwarz function of the wave, and the work builds on a novel framework set out recently by Crowdy (J. Fluid Mech., vol. 954, 2022, A47). Conformal mapping theory is used to construct Schwarz functions with the requisite properties and to parametrize the waveform. A two-parameter family of solutions is found by solving a pair of nonlinear algebraic equations. This system of equations has intriguing properties: indeed, it is degenerate, which radically reduces the number of possible solutions, although the space of physically admissible equilibria is still found to be rich and diverse. For inline vortex streets, where the two vortex rows are aligned vertically, there is generally a single physically admissible solution. However, for staggered streets, where the two vortex rows are offset horizontally, certain parameter regimes produce multiple solutions. An important outcome of the work is that while only degenerate von Kármán point vortex streets can exist in an unbounded simple shear current, a broad array of such equilibria is possible in a shear current beneath a cotravelling wave on a free surface.

Discrete vortex rings impinging on concave hemispherical cavities were explored experimentally. Planar laser-induced fluorescence, two-dimensional particle image velocimetry and flow visualization techniques were employed. Five different ratios of vortex ring to hemisphere cavity radius ($\gamma$) were investigated, namely, $\gamma = 1/4,1/3,2/5,$$1/2, 2/3$. For $\gamma = 1/4,1/3, 2/5$, the geometric confinement of the primary ring due to the hemispherical cavity induced loop-like instabilities in the secondary ring, which led to head-on collision and ejection of the looped ends as they orbited the primary ring. As the hemispherical cavity decreased in diameter (increasing $\gamma$), the dynamics were altered significantly due to the increased generation of vorticity along the edge of the hemisphere. For $\gamma = 1/2$, vorticity produced at the edge/lip of the hemisphere ultimately disrupted the classical formation of a secondary vortex ring from the wall-bounded vorticity. For $\gamma = 2/3$, the primary ring and hemisphere radius were close enough in size that the interaction was dominated by direct impact of the primary ring with the lip of the cavity. The primary vortex ring produced a vortex ring at the lip of the hemisphere that ultimately separated from the cavity, orbited around the primary ring, and then self-advected in the direction opposite to the primary vortex ring trajectory. A detailed investigation of the dynamics provided.

This paper presents the effects of wing kinematics on both normal forward flight and escape flight of a dragonfly. A Navier–Stokes-based numerical model has been adopted, and results have been substantiated by experimental data. The wing kinematics of tethered specimens and the prescribed wing morphology of a free-flying dragonfly were used in the simulation. To shed light on the interplay between kinematics and aerodynamics, a parametric study of the kinematics has been conducted. It is found that in escape flight, the dragonfly generates additional lift while the thrust reduces and the overall efficiency drops. Compared with normal forward flight, the escape mode produces larger lift force peaks. When the kinematics change to facilitate escape flight, the aerodynamic forces are affected by not only the flapping kinematics but, in the case of the hindwing, the varied wing–wing vortex interactions. The direction of the resultant force on each wing changes according to the change of the mean of pitching angle and stroke plane angle. We found that in the studied configurations, the varied phasing of the wings has a marginal effect on the aerodynamics of the dragonfly. It reduces lift and increases thrust, and this force modulation is slightly more efficient when the local angle of attack also changes. On the other hand, the change of angle of attack played a major role in leading-edge vortex formations and directing the resultant forces of the wings. The results can be useful in developing flight control strategies for micro air vehicle design.