Low-frequency phenomena in an incompressible pressure-induced laminar separation bubble (LSB) on a flat plate is investigated using direct numerical simulation. The LSB configuration of Spalart and Strelets (J. Fluid Mech., vol. 403, 2000, pp. 329–349) is used. Wall pressure spectra indicate low-frequency-flapping $(St \sim 0.08)$ and high-frequency-shedding $(St \sim 1.52)$ regimes. Conditional velocity averages based on the fraction of reversed flow reveal the low frequency as an expansion/contraction of the LSB. While the high frequency only exhibits exponential growth within the LSB up to breakdown of the spanwise rollers, the low frequency and velocity fluctuations exhibit exponential growth upstream of separation. Instantaneous flow fields reveal large streamwise streaky structures forming within the LSB and extending past reattachment, much like high and low speed streaks in turbulent boundary layers. A predominance of sweep-like events ($Q4$) is observed during contraction and of ejection-like events ($Q2$) during expansion. These motions appear as dominant low-frequency modes in three-dimensional proper orthogonal and dynamic mode decompositions, exhibiting spatial amplification from separation to reattachment. The advection of a group of spanwise alternating streaky structures past the LSB results in an overall contraction after which the bubble expands to its ‘unforced’ state in the absence of the streaks. The low frequency then corresponds to the time it takes for streaks to form, amplify and advect past the LSB from separation to reattachment. This behaviour is linked to the mean flow deformation reported by Marxen and Rist (J. Fluid Mech., vol. 660, 2010, pp. 37–54), where the presence of streaks results in reduced mean bubble size. The formation of these streaky structures, in the absence of free stream turbulence, may be attributed to an absolute instability of the LSB due to the development of a secondary bubble within the primary.

The spatio-temporal scales, as well as a comprehensive self-sustained mechanism of the reattachment unsteadiness in shock wave/boundary layer interaction, are investigated in this study. Direct numerical simulations reveal that the reattachment unsteadiness of a Mach 7.7 laminar inflow causes over $26\,\%$ variation in wall friction and up to $20\,\%$ fluctuation in heat flux at the reattachment of the separation bubble. A statistical approach, based on the local reattachment upstream movement, is proposed to identify the spanwise and temporal scales of reattachment unsteadiness. It is found that two different types, i.e. self-induced and random processes, dominate different regions of reattachment. A self-sustained mechanism is proposed to comprehend the reattachment unsteadiness in the self-induced region. The intrinsic instability of the separation bubble transports vorticity downstream, resulting in an inhomogeneous reattachment line, which gives rise to baroclinic production of quasi-streamwise vortices. The pairing of these vortices initiates high-speed streaks and shifts the reattachment line upstream. Ultimately, viscosity dissipates the vortices, triggering instability and a new cycle of reattachment unsteadiness. The temporal scale and maximum vorticity are estimated with the self-sustained mechanism via order-of-magnitude analysis of the enstrophy. The advection speed of friction, derived from the assumption of coherent structures advecting with a Blasius-type boundary layer, aligns with the numerical findings.

We investigate effect of porous insert located upstream of the separation edge of a backward-facing step (BFS) in early transitional regime as a function of Reynolds number. This is an example of hydrodynamic system that is a combination of separated shear flow with large amplification potential and porous materials known for efficient flow destabilisation. Spectral analysis reveals that dynamics of BFS is dominated by spectral modes that remain globally coherent along the streamwise direction. We detect two branches of characteristic frequencies in the flow and with Hilbert transform we characterise their spatial support. For low Reynolds numbers, the dynamics of the flow is dominated by lower frequency, whereas for sufficiently large Reynolds numbers cross-over to higher frequencies is observed. Increasing permeability of the porous insert results in decrease in Reynolds number value, at which frequency cross-over occurs. By comparing normalised frequencies on each branch with local stability analysis, we attribute Kelvin–Helmholtz and Tollmien–Schlichting instabilities to upper and lower frequency branches, respectively. Finally, our results show that porous inserts enhance Kelvin–Helmholtz instability and promote transition to oscillator-type dynamics. Specifically, the amplitude of vortical (BFS) structures associated with higher-frequency branch follows Landau model prediction for all investigated porous inserts.

The turbulent wake behind a flat-back Ahmed body is investigated using stacked stereoscopic particle image velocimetry. The wake is disturbed by a steady jet from the centre of the base and the effects are quantified for key blowing rates. The unactuated wake exhibits bistable dynamics in the horizontal plane that are completely subdued for the optimal blowing case, yielding a base drag reduction of 9 %. The three-dimensional mean wake is reconstructed and used to evaluate the wake mass fluxes whose equilibrium determines the recirculation length. The results for the unactuated wake show that up to 80 % of replenishment fluid flux entering the recirculation bubble from the free-stream flow is provided through the low-pressure side of the base, where the symmetry-breaking shear layer roll-up occurs near the base. For the optimal blowing configuration, where the wake becomes symmetric, the flux of wake replenishment is severely reduced. This flow configuration results in elongated shear layers on all sides, which terminate the bubble with a roll-up of reduced intensity at a further downstream location. The dominant cause of bubble growth and the accompanying drag reduction is attributed to the momentum of the base blowing, and the new regime is referred to as the ‘favourable momentum regime’. Similar trends are observed when the model is at $5^{\circ }$ yaw where a reduction of drag and yaw-induced asymmetry are obtained. Proper orthogonal decomposition of the wake reveals the coherent structures related to the bistable flow and the symmetric wake under optimal blowing coefficient.

High angle of attack flows over swept three-dimensional wings based on the NACA 0015 profile are studied numerically at low Reynolds numbers. Linear stability analysis is used to compute instability and receptivity of the flow via the respective three-dimensional (triglobal) direct and adjoint eigenmodes. The magnitude of the adjoint eigenvectors is used to identify regions of maximum flow receptivity to momentum forcing. It is found that such regions are located above the primary three-dimensional separation line, their spanwise position varying with wing sweep. The wavemaker region corresponding to the leading global eigenmode is computed and found to lie inside the laminar separation bubble (LSB) at the spanwise location of peak recirculation. Increasing the Reynolds number leads to the wavemaker becoming more compact in the spanwise direction, and concentrated in the top and bottom shear layers of the LSB. As sweep is introduced, the wavemaker moves towards the wing tip, following the spanwise displacement of maximum recirculation. Flow modifications resulting from application of different types of forcing are studied by direct numerical simulation initialised with insights gained from stability analysis. Periodic forcing at the regions of maximum receptivity to momentum forcing results in greater departure from the baseline case compared to same (low, linear) amplitude forcing applied elsewhere, underlining the potential of linear stability analysis to identify optimal regions for actuator positioning.

Dynamic stall at low Reynolds numbers, $Re \sim O(10^4)$, exhibits complex flow physics with co-existing laminar, transitional and turbulent flow regions. Current state-of-the-art stall onset criteria use parameters that rely on flow properties integrated around the leading edge. These include the leading edge suction parameter or $LESP$ (Ramesh et al., J. Fluid Mech., vol. 751, 2014, pp. 500–538) and boundary enstrophy flux or $BEF$ (Sudharsan et al., J. Fluid Mech., vol. 935, 2022, A10), which have been found to be effective for predicting stall onset at moderate to high $Re$. However, low-$Re$ flows feature strong vortex-shedding events occurring across the entire airfoil surface, including regions away from the leading edge, altering the flow field and influencing the onset of stall. In the present work, the ability of these stall criteria to effectively capture and localize these vortex shedding events in space and time is investigated. High-resolution large-eddy simulations for an SD7003 airfoil undergoing a constant-rate, pitch-up motion at two $Re$ (10 000 and 60 000) and two pitch rates reveal a rich variety of unsteady flow phenomena, including instabilities, transition, vortex formation, merging and shedding, which are described in detail. While stall onset is reflected in both $LESP$ and $BEF$, local vortex-shedding events are identified only by the $BEF$. Therefore, $BEF$ can be used to identify both dynamic stall onset and local vortex-shedding events in space and time.

We perform direct numerical simulations of actively controlled laminar separated wakes around low-aspect-ratio wings with two primary goals: (i) reducing the size of the separation bubble and (ii) attenuating the wing tip vortex. Instead of preventing separation, we modify the three-dimensional (3-D) dynamics to exploit wake vortices for aerodynamic enhancements. A direct wake modification is considered using optimal harmonic forcing modes from triglobal resolvent analysis. For this study, we consider wings at angles of attack of $14^\circ$ and $22^\circ$, taper ratios $0.27$ and $1$, and leading edge sweep angles of $0^\circ$ and $30^\circ$, at a mean-chord-based Reynolds number of $600$. The wakes behind these wings exhibit 3-D reversed-flow bubble and large-scale vortical structures. For tapered swept wings, the diversity of wake vortices increases substantially, posing a challenge for flow control. To achieve the first control objective for an untapered unswept wing, root-based actuation at the shedding frequency is introduced to reduce the reversed-flow bubble size by taking advantage of the wake vortices to significantly enhance the aerodynamic performance of the wing. For both untapered and tapered swept wings, root-based actuation modifies the stalled flow, reduces the reversed-flow region and enhances aerodynamic performance by increasing the root contribution to lift. For the goal of controlling the tip vortex, we demonstrate the effectiveness of actuation with high-frequency perturbations near the tip. This study shows how insights from resolvent analysis for unsteady actuation can enable global modification of 3-D separated wakes and achieve improved aerodynamics of wings.

During stroke reversals, insect wings interact with their own wake flow from the preceding half-stroke, resulting in an unsteady aerodynamic mechanism known as ‘wing–wake interaction’ or ‘wake capture’. To better elucidate this mechanism, we numerically solved the incompressible Navier–Stokes equations at Reynolds numbers $10^2$ and $10^3$. Simulations were conducted for wing planforms defined using the beta function distribution with varying aspect ratios ($AR=2\unicode{x2013}6$) and radial centroid locations ($\hat {r}_1=0.4\unicode{x2013}0.6$), whilst employing representative normal hovering kinematics. The wake development from the considered flapping wing planforms was investigated, and the wake capture contribution to aerodynamic force production was quantified by comparing the force generation between the fifth and first stroke cycles at multiple sections along the wingspan. Our results revealed that on the inboard wing region experiencing an attached leading-edge vortex (LEV) structure, wing–wake interaction is dominated by an unsteady downwash effect, resulting in a reduction in local force production. However, in regions closer to the wingtip experiencing detachment of the LEV, wing–wake interaction is dominated by an unsteady upwash effect, leading to an increase in local force production. Consequently, the global wake capture force production is controlled by the extent of LEV detachment, which primarily increases with the increase of wing aspect ratio. This suggests that for normal hovering flapping wings, the typical loss in translational force production due to wingtip stall is partially mitigated by wake capture effects.

At high incidence, low-aspect-ratio wings present a unique set of aerodynamic characteristics, including flow separation, vortex shedding and unsteady force production. Furthermore, low-aspect-ratio wings exhibit a highly impactful tip vortex, which introduces strong spanwise gradients into an already complex flow. In this work, we explore the interaction between leading-edge flow separation and a strong, persistent tip vortex over a Reynolds number range of $600 \leq Re \leq 10{\,}000$. In performing this study, we aim to bridge the insight gained from existing low-Reynolds-number studies of separated flow on finite wings ($Re \approx 10^2$) and turbulent flows at higher Reynolds numbers ($Re \approx 10^4$). Our study suggests two primary effects of the Reynolds number. First, we observe a break from periodicity, along with a dramatic increase in the intensity and concentration of small-scale eddies, as we shift from $Re = 600$ to $Re = 2500$. Second, we observe that many of our flow diagnostics, including the time-averaged aerodynamic force, exhibit reduced sensitivity to Reynolds number beyond $Re = 2500$, an observation attributed to the stabilising impact of the wing tip vortex. This latter point illustrates the manner by which the tip vortex drives flow over low-aspect-ratio wings, and provides insight into how our existing understanding of this flow field may be adjusted for higher-Reynolds-number applications.

Comprehensive coherent structures around a surface-mounted low aspect ratio square cylinder in uniform flow with an oblique angle of $45^{\circ }$ were investigated for cylinder-width-based Reynolds numbers of 3000 and 10 000 by direct numerical simulation based on a topology-confined mesh refinement framework. High-resolution simulations and the critical-point concept were scrutinized to reveal for the first time the reasonable and compatible topologies of flow separation and complete near-wall structures, due to their extensive impact on various engineering applications. Large-scale horseshoe vortices are observed at two notable foci in the viscous sublayer. Within this layer, a wall-parallel jet is formed by downflow intruding into the bottom surface at the half-saddle point, then deflecting in the upstream direction and finally penetrating the bottom surface until the half-saddle point. A pair of conical vortices on the cylinder's top surface switch themselves on two sides of the square cylinder, where the switching frequency is identical with that of the sway of the side shear layer. The undulation of the Kelvin–Helmholtz instability is identified in the instantaneous development of a conical vortex and side shear layer, where the scaling of the ratio of the Kelvin–Helmholtz and von Kármán frequencies follows the power-law relation obtained by Lander et al. (J. Fluid Mech., vol. 849, 2018, pp. 1096–1119). Large-scale arch-shaped vortex is often detected in the intermediate wake region of a square cylinder, involving two interconnected portions, such as the leg portion separated from leeward surfaces and head portion rolled up from the top surface. The leg portion of the arch-shaped vortex was rooted by two foci near the bottom-surface plane.

The low-frequency modal and non-modal linear dynamics of an incompressible, pressure-gradient-induced turbulent separation bubble (TSB) are investigated, with the objective of studying the mechanism responsible for the low-frequency contraction and expansion (breathing) commonly observed in experimental studies. The configuration of interest is a TSB generated on a flat test surface by a succession of adverse and favourable pressure gradients. The base flow selected for the analysis is the average TSB from the direct numerical simulation of Coleman et al. (J. Fluid Mech., vol. 847, 2018, pp. 28–70). Global mode analysis reveals that the eigenmodes of the linear operator are damped for all frequencies and wavenumbers. Furthermore, the least damped eigenmode appears to occur at zero frequency and low, non-zero spanwise wavenumber when scaled with the separation length. Resolvent analysis is then employed to examine the forced dynamics of the flow. At low frequency, a region of low, non-zero spanwise wavenumber is also discernible, where the receptivity appears to be driven by the identified weakly damped global mode. The corresponding optimal energy gain is shown to have the shape of a first-order, low-pass filter with a cut-off frequency consistent with the low-frequency unsteadiness in TSBs. The results from resolvent analysis are compared to the unsteady experimental database of Le Floc'h et al. (J. Fluid Mech., vol. 902, 2020, A13) in a similar TSB flow. The alignment between the optimal response and the first spectral proper orthogonal decomposition mode computed from the experiments is shown to be close to $95\,\%$, while the spanwise wavenumber of the optimal response is consistent with that of the low-frequency breathing motion captured experimentally. This indicates that the fluctuations observed experimentally at low frequency closely match the response computed from resolvent analysis. Based on these results, we propose that the forced dynamics of the flow, driven by the weakly damped global mode, serve as a plausible mechanism for the origin of the low-frequency breathing motion commonly observed in experimental studies of TSBs.

Motivated by contradicting or insufficient information regarding the large-scale flow dynamics around surface-mounted finite-height square prisms of small aspect ratio, the present study investigates the dominant vortex shedding and low-frequency dynamics around a surface-mounted cube. These flow modes were obtained from the spectral proper orthogonal decomposition of large-eddy simulation results, at a Reynolds number of $\textit {Re}=1\times 10^4$ and two different types of boundary layer: a thin and laminar boundary layer with thickness $\delta /D=0.2$ and a thick and turbulent boundary layer with $\delta /D=0.8$. The main antisymmetric mode pair revealed a new flow pattern with the alternate shedding of streamwise flow structures, indicating a transition from the half-loops of taller prisms to only streamwise strands (i.e. no vertical core) for smaller aspect ratio. The formation process of the streamwise structures is due to a reorientation of the vorticity of the arch vortex in the streamwise direction characteristic of the shed structures. The low-frequency drift mode affected the length of the recirculation region, the strength of vortex shedding, and the near-wall flow field and pressure distribution on the cube's faces, leading to low-frequency variations in the fluctuating drag and normal force coefficients. These large-scale flow dynamics were similar for both boundary layers, but minor differences were identified, related mostly to the occurrence of flow attachment and the formation of a headband vortex for the thicker boundary layer.

Previous studies have shown that low-frequency vortex oscillations occur around a hemisphere–cylinder body at different angles of attack, but the underlying mechanism is still unclear. In this study, we examine the origin of the vortex oscillation using numerical simulations and global linear stability analysis. The vortex oscillation is reproduced using numerical simulations, and the oscillatory modes are computed through dynamic mode decomposition (DMD). We obtain the base flow through a selective frequency damping method, which exhibits a pair of steady leeward vortices over the body. The four unstable modes are computed using a modified Arnoldi iteration. The antisymmetric mode with a Strouhal number of 0.105 is discovered to be responsible for the alternate oscillation of the vortex pair, and the mode with a Strouhal number of 0.220 corresponds to the in-phase vortex oscillation. Their frequencies have good agreement with the modes of DMD. The other two unstable modes with higher frequencies, one antisymmetric and one symmetric, are harmonic frequencies of the above two modes. The study conclusively verifies that the vortex oscillation over a hemisphere–cylinder body originates from a global flow instability.

The transient processes of laminar separation bubble (LSB) formation and bursting on a rectangular NACA 0018 wing are studied experimentally. A two-dimensional airfoil model is used as a baseline for the assessment of finite wing effects. The models are subjected to ramp changes in free-stream velocity causing the flow to switch between a state where an LSB forms and a state without reattachment. Lift force and particle image velocimetry measurements are used to relate the flow development to the aerodynamic loading. The lift coefficient of the airfoil exhibits substantial hysteresis, and the duration of the lift transients range from $10$ to $22$ convective time scales for bubble formation and $22$ to $30$ convective time scales for bursting. In contrast, the transient lift coefficients of the wing change gradually, with less hysteresis. The wing tip causes greater three-dimensionality in the separation bubble, whose thickness increases near the midspan where bursting is initiated. During bubble formation, the region of separated flow contracts towards the midspan. The gradual change in lift of the wing is linked to slower spanwise expansion and contraction of the separated flow region relative to the airfoil. On both models, the wavenumbers and amplitudes of disturbances in the separated shear layer rapidly change when reattachment initiates or ceases. Applying the bursting criterion of Gaster (Tech. Rep. Reports and Memoranda 3595. Aeronautical Research Council, London, 1967) to the bubble on the wing shows that bursting of the bubble at a single spanwise location is insufficient to cause complete spanwise failure of reattachment, and that the relationship between bursting parameters depends on spanwise position.

This experimental study investigates the fluid–structure–surface interactions of a flexibly mounted rigid plate in axial flow, focusing on flow-induced vibration (FIV) response and vortex dynamics of the system within a reduced velocity range of $U^*=0.29\unicode{x2013}8.73$, corresponding to a Reynolds number range of $Re=518\unicode{x2013}15\,331$. The plate, with one and two degrees of freedom (DoFs) for pitching and plunging oscillations, is examined at various submerged heights near the free surface. Results show that the plate exhibits divergence instability at low reduced velocities in both 1DoF and 2DoF systems. As the flow velocity surpasses a critical reduced velocity, periodic limit-cycle oscillations (LCOs) occur, increasing in amplitude until a second critical reduced velocity is reached. Beyond this point, LCOs are suppressed, and the plate experiences an increased static divergence angle with further flow velocity increase. The proximity to the free surface significantly influences the FIV response, with decreasing submerged heights leading to reduced LCO amplitudes and a shift of instabilities to higher reduced velocities. Vortex dynamics are analysed using time-resolved volumetric particle tracking velocimetry and hydrogen bubble flow visualisation. The analysis reveals disruptions in the symmetric flow field near the free surface, causing elongation and fragmentation of vortices in the wake of the plate, as well as vortex coupling. Proper orthogonal decomposition (POD) identifies dominant coherent structures, including leading-edge and trailing-edge vortices, captured in the first and second paired modes. On the other hand, higher POD modes capture the interaction of vortices in the wake and near the free surface.

A study of the physics of separating and reattaching flows around bodies with sharp edges is reported. Data from direct numerical simulations of the flow around a rectangular cylinder with aspect ratio 5 at different Reynolds numbers are used. The flow is decomposed into multiple interacting flow phenomena such as the laminar boundary layer in the front face, the separated shear layer, the flow impingement at reattachment, the reverse boundary layer within the recirculating bubble and the near- and far-wake flow. A detailed analysis of the physics of these phenomena is provided, including the slow modulation induced by large-scale instabilities related with vortex shedding. The entrainment phenomena acting along the separated shear layer and their unbalance between its inner and outer sides are recognised as fundamental mechanisms determining the tendency of the flow to reattach and the overall fluxes of momentum and heat. The behaviour of entrainment is found to be strictly related with the shear-layer velocity difference that in turn is determined by the behaviour of the reverse boundary layer and by its strength in counteract adverse pressure gradients. The physical understanding of the compound role played by these and all the other mechanisms composing the flow, poses the basis for the formulation of theoretical frameworks able to unify all these interacting phenomena. Finally, the present work provides access to high-fidelity flow statistics of relevance for benchmark activities on bluff bodies with sharp edges.

Three-dimensional flows over low-aspect-ratio rectangular flat plates (${A{\kern-4pt}R} = 1.00$–$1.50$) are investigated using tomographic and planar particle image velocimetry techniques. The chord-based Reynolds number is $5400$, and the angle of attack is fixed at $6^\circ$. This study reveals for the first time the interplay between spanwise fluid transport and downwash, both originating from the tip effects. Spanwise fluid transport promotes the formation and subsequent coherent development of leading-edge vortices, whereas downwash stabilizes the flow. Specifically, two mechanisms related to spanwise fluid transport are revealed. First, the spanwise fluid transport enhances the intensity of the reversed flow, promoting the shear layer roll-up and vortex shedding. Second, the near-wall spanwise flow interacts with the shed C-shape vortices, thereby strengthening the vortex heads. In particular, through these interactions, spanwise fluid transport can sustain the coherence of the C-shape vortices until the vortex heads split in a regular fashion. Consequently, the C-shape vortices are transformed into novel Þ-shape vortices for the plates of ${A{\kern-4pt}R} \leq 1.25$, which supplements the previously discovered transformation from C-shape to M-shape vortices for larger ${A{\kern-4pt}R}$ plates. Downstream of this novel vortex-splitting transformation, two fundamental processes contribute to the formation of hairpin vortices. The above comprehensive understanding of complete vortex evolution routine provides valuable insights into the tip effects on the formation of three-dimensional flows over low-${A{\kern-4pt}R}$ plates.

Supersonic backward-facing step (BFS) flow is numerically studied using direct numerical simulation (DNS) and global stability analysis (GSA) with a free stream Mach number of 2.16 and a Reynolds number of 7.938 × 105 based on the flat-plate length L and free stream conditions. Two-dimensional BFS flow becomes unstable to three-dimensional perturbations as the step height h exceeds a certain value, while no two-dimensionally unstable mode is found. Global instability occurs with the fragmentation of the primary separation vortex downstream of the step. Two stationary modes and one oscillatory unstable mode are obtained at a supercritical ratio of L/h = 32.14, among which the two stationary modes originate from the coalescence of a pair of conjugate modes. The most unstable mode manifests itself as streamwise streaks in the reattached boundary layer, which is similar to that in shock-induced separated flow, although the flow separation mechanisms are different. Without introducing any external disturbances, the DNS captures the preferred perturbations and produces a growth rate in agreement with the GSA prediction in the linear growth stage. In the quasi-steady stage, the secondary separation vortex breaks up into several small bubbles, and the number of streamwise streaks is doubled. A low-frequency unsteadiness that may be associated with the oscillatory mode is also present.

Flow through sudden expansion finds its application in several engineering and biological processes. Though the stability of flow through steady sudden expansion has garnered much attention, little to none is given to the pulsatile flow through sudden expansion. Hence, in the present work we study the influence of inflow pulsatility on flow characteristics in a sudden expansion. The inflow velocity is a sinusoidal waveform that is modulated to encompass a wide range of amplitudes, ${{a}}$, and reduced velocities, ${{U_{r}}}$. We report four different modes, namely, synchronized growth of the recirculation region (at high ${{U_{r}}}$), necking and diffusion of the recirculation region (at moderately high ${{U_{r}}}$), splitting and convection of the recirculation region (at moderate ${{U_{r}}}$) and inverse growth of the recirculation region (at low ${{U_{r}}}$). In each mode, the symmetry-breaking critical Reynolds number is obtained through numerical experiments and compared with those of Floquet stability analysis. We found that diffusion and the convection mode of the recirculation region increases the stability of the flow while the inverse growth mode of the recirculation region decreases the same. The effect of the expansion ratio, ${{ER}}$, is also explored, and we found that as ${{ER}}$ increases, the absolute stability of flow decreases, but relative stability between the modes remains similar. Finally, we explain the dynamics of the modes by using terms involving the vorticity transport equation.

A low-order vortex model has been developed for analysing the unsteady aerodynamics of airfoils. The model employs an infinitely thin vortex sheet in place of the attached boundary layer and a sheet of point vortices for the shed shear layer. The strength and direction of the vortex sheet shed at the airfoil trailing edge are determined by an unsteady Kutta condition. The roll-up of the ambient shear layer is represented by a unique point vortex, which is consistently fed circulation by the last point vortex of the free vortex sheet. The model's dimensionality is reduced by using three tuning parameters to balance representational accuracy and computational efficiency. The performance of the model is evaluated through experiments involving impulsively started and heaving and pitching airfoils. The model accurately captures the dynamics of the development and evolution of the shed vortical structure while requiring minimal computational resources. The validity of the model is confirmed through comparison with experimental force measurements and a baseline unsteady panel method that does not transfer circulation in the free vortex sheet.