Reynolds number, thickness and camber effects on flapping airfoil propulsion

The effect of varying airfoil thickness and camber on plunging and combined pitching and plunging airfoil propulsion at Reynolds number Re=200, 2000, 20 000 and 2×10⁶ was studied by numerical simulations for fully laminar and fully turbulent flow regimes. The thickness study was performed on 2-D NACA symmetric airfoils with 6-50% thick sections undergoing pure plunging motion at reduced frequency k=2 and amplitudes h=0.25 and 0.5, and for combined pitching and plunging motion at k=2, h=0.5, phase ?=90°, pitch angle θ_o=15° and 30° and the pitch axis was located at 1/3 of chord from leading edge. At Re=200 for motions where positive thrust is generated, thin airfoils outperform thick airfoils. At higher Re significant gains could be achieved both in thrust generation and propulsive efficiency by using a thicker airfoil section for plunging and combined motion with low pitch amplitude. The camber study was performed on 2-D NACA airfoils with varying camber locations undergoing pure plunging motion at k=2, h=0.5 and Re=20 000. Little variation in thrust performance was found with camber. The underlying physics behind the alteration in propulsive performance between low and high Reynolds numbers has been explored by comparing viscous Navier-Stokes and inviscid panel method results. The role of leading edge vortices was found to be key to the observed performance variation.     

Thrust augmentation of flapping airfoils in low Reynolds number flow using a flexible membrane

The unsteady aerodynamic thrust and aeroelastic response of a two-dimensional membrane airfoil under prescribed harmonic motion are investigated computationally with a high-order Navier–Stokes solver coupled to a nonlinear membrane structural model. The effects of membrane prestress and elasticity are examined parametrically for selected plunge and pitch–plunge motions at a chord-based Reynolds number of 2500. The importance of inertial membrane loads resulting from the prescribed flapping is also assessed for pure plunging motions. This study compares the period-averaged aerodynamic loads of flexible versus rigid membrane airfoils and highlights the vortex structures and salient fluid–membrane interactions that enable more efficient flapping thrust production in low Reynolds number flows.     

Parameter dependence of vortex interactions on a two-dimensional plunging plate

The structure and dynamics of the flow field created by a plunging flat-plate airfoil are investigated at a chord Reynolds number of 10,000 while varying plunge amplitude and Strouhal number. Digital particle image velocimetry measurements are used to characterize the shedding patterns and the interactions between the leading- and trailing-edge vortex structures (LEV and TEV), resulting in the development of a wake classification system based on the nature and timing of interactions between the leading- and trailing-edge vortices. The streamwise advancement of the LEV during a plunge cycle and its resulting interaction with the TEV is primarily dependent on reduced frequency; however, for Strouhal numbers above approximately 0.4, significant changes are observed in the formation of vortices shed from the leading and trailing edges, as well as the circulation of the leading-edge vortex. The functional form of the relationship between leading-edge vortex circulation and Strouhal number suggests that the Strouhal number dependence is more specifically a manifestation of the effective angle of attack. Comparison with low-Reynolds-number studies of plunging airfoil aerodynamics reveals a high degree of consistency and suggests applicability of the classification system beyond the range examined in the present work.     

Effect of flexure on aerodynamic propulsive efficiency of flapping flexible airfoil

The aim of present study is to investigate the effect of chord-wise flexure amplitude on unsteady aerodynamic characteristics for a flapping airfoil with various combinations of Reynolds number and reduced frequency. Unsteady, viscous flows over a single flexible airfoil in plunge motion are computed using conformal hybrid meshes. The dynamic mesh technique is applied to illustrate the deformation modes of the flexible flapping airfoil. In order to investigate the influence of the flexure amplitude on the aerodynamic performance of the flapping airfoil, the present study considers eight different flexure amplitudes (a₀) ranging from 0 to 0.7 in intervals of 0.1 under conditions of Re=10⁴, reduced frequency k=2, and dimensionless plunge amplitude h₀=0.4. The computed unsteady flow fields clearly reveal the formation and evolution of a pair of leading edge vortices along the body of the flexible airfoil as it undergoes plunge motion. Thrust-indicative wake structures are generated when the flexure amplitude of the airfoil is less than 0.5 of the chord length. An enhancement in the propulsive efficiency is observed for a flapping airfoil with flexure amplitude of 0.3 of the chord length. This study also calculates the propulsive efficiency and thrust under various Reynolds numbers and reduced frequency conditions. The results indicate that the propulsive efficiency has a strong correlation with the reduced frequency. It is found that the flow conditions which yield the highest propulsive efficiency correspond to Strouhal number St of 0.255.     

Effects of a dynamic trailing-edge flap on the aerodynamic performance and flow structures in hovering flight

To examine the effects of wing morphing on unsteady aerodynamics, deformable flapping plates are numerically studied in a low-Reynolds-number flow. Simulations are carried out using an in-house immersed-boundary-method-based direct numerical simulation (DNS) solver. In current work, chord-wise camber is modeled by a hinge connecting two rigid components. The leading portion is driven by a biological hovering motion along a horizontal stroke plane. The hinged trailing-edge flap (TEF) is controlled by a prescribed harmonic deflection motion. The effects of TEF deflection amplitude, deflection phase difference, hinge location, and Reynolds number on the aerodynamic performance and flow structures are investigated. The results show that the unsteady aerodynamic performance of deformable flapping plates is dominated by the TEF deflection phase difference, which directly affects the strength of the leading-edge vortex (LEV) and thus influences the entire vortex shedding process. The overall lift enhancement can reach up to 26% by tailoring the deflection amplitude and deflection phase difference. It is also found that the role of the dynamic TEF played in the flapping flight is consistent over a range of hinge locations and Reynolds numbers. Results from a low aspect-ratio (AR=2) deformable plate show the same trend as those of 2-D cases despite the effect of the three-dimensionality.     

Unsteady airfoil with a harmonically deflected trailing-edge flap

The effects of a harmonically deflected trailing-edge flap, actuated at different start times and amplitudes but with frequency different from the airfoil motion, on the aerodynamic loads of an oscillating NACA 0015 airfoil were investigated experimentally at Re=2.51×10⁵. Both in-phase and 180° out-of-phase flap deflections, relative to the airfoil motion, were tested. The results show that there was a large change in the hysteretic behavior of the dynamic load loops, and that the formation and detachment of the leading-edge vortex (LEV) were not affected by the flap motion, while the low pressure signature of the vortex was affected by the flap actuation start time. The later the flap actuation the larger the change in the strength of the LEV. The present flap control scheme was also found to be as effective as that achieved by a pulsed ramp flap motion, but with a reduced number of control parameters.     

Lift enhancement through flexibility of plunging wings at low Reynolds numbers

In this paper the combined effect of two mechanisms for lift enhancement at low Reynolds numbers are considered, wing oscillations and wing flexibility. The force, deformation and flow fields of rigid and flexible low aspect ratio (AR=3) and high aspect ratio (AR=6) wings oscillating at a fixed post-stall angle of attack of 15° and amplitude of 15% of chord are measured. The force measurements show that flexibility can increase the time-averaged lift coefficient significantly. For low aspect ratio wings the maximum lift coefficient across all Strouhal numbers was C_l=1.38 for the rigid wing as opposed to C_l=2.77 for the flexible wing. Very similar trends were observed for the high aspect ratio wings. This increase is associated with significant deformation of the wing. The root is sinusoidally plunged with small amplitude but this motion is amplified along the span resulting in a larger tip motion but with a phase lag. The amount it is amplified strongly depends on Strouhal number. A Strouhal number of Sr_c=1.5 was selected for detailed flow field measurements due to it being central to the high-lift region of the flexible wings, producing approximately double the lift of the rigid wing. For this Strouhal number the rigid wings exhibit a Leading Edge Vortex (LEV) ring. This is where the clockwise upper-surface LEV pairs with the counter-clockwise lower-surface LEV to form a vortex ring that self-advects upstream and away from the wing's upper surface. Conversely the deformation of the flexible wings inhibits the formation of the LEV ring. Instead a strong upper-surface LEV forms during the downward motion and convects close to the airfoil upper surface thus explaining the significantly higher lift. These measurements demonstrate the significant gains that can be achieved through the combination of unsteady aerodynamics with flexible structures.     

Low Reynolds number flow over a square cylinder with a detached flat plate

A numerical study of the alteration of a square cylinder wake using a detached downstream thin flat plate is presented. The wake is generated by a uniform flow of Reynolds number 150 based on the side length of the cylinder, D. The sensitivity of the near wake structure to the downstream position of the plate is investigated by varying the gap distance (G) along the wake centerline in the range 0 ⩽ G ⩽ 7D for a constant plate length of L = D. A critical gap distance is observed to occur at G_c ≈ 2.3D that indicates the existence of two flow regimes. Regime I is characterised by vortex formation occurring downstream of the gap while for regime II, formation occurs within the gap. By varying the plate length and gap distance, a condition is found where significant unsteady total lift reduction can occur. The root mean square lift reduction is limited by an unsteady stall process on the plate.     

Two-dimensional unsteady flow around a waving plate in wall effects

A new method of nonlinear formulation is presented to analyze the two-dimensional incompressible flow around a flexible plate waving near a rigid wall. A system of differential and integral equations is solved for the velocity field and the wake vortex. A nonlinear unsteady Kutta condition is imposed at the trailing edge in order to treat the case of large amplitude and fast oscillation accurately. The shed vortex sheet is discretized and approximated by a large number of vortex filaments, and their movements are visualized by numerical computation. The lift, thrust, power input and hydrodynamic efficiency are computed for various values of the distance of the waving plate from the wall.     

Flow field characteristics study of a flapping airfoil using computational fluid dynamics

The flow field of a flapping airfoil in Low Reynolds Number (LRN) flow regime is associated with complex nonlinear vortex shedding and viscous phenomena. The respective fluid dynamics of such a flow is investigated here through Computational Fluid Dynamics (CFD) based on the Finite Volume Method (FVM). The governing equations are the unsteady, incompressible two-dimensional Navier-Stokes (N-S) equations. The airfoil is a thin ellipsoidal geometry performing a modified figure-of-eight-like flapping pattern. The flow field and vortical patterns around the airfoil are examined in detail, and the effects of several unsteady flow and system parameters on the flow characteristics are explored. The investigated parameters are the amplitude of pitching oscillations, phase angle between pitching and plunging motions, mean angle of attack, Reynolds number (Re), Strouhal number (St) based on the translational amplitudes of oscillations, and the pitching axis location (x/c). It is shown that these parameters change the instantaneous force coefficients quantitatively and qualitatively. It is also observed that the strength, interaction, and convection of the vortical structures surrounding the airfoil are significantly affected by the variations of these parameters.     

Effects of large spanwise wavelength on the wake of a sinusoidal wavy cylinder

The wake of a sinusoidal wavy cylinder with a large spanwise wavelength λ/D_m (=3.79–7.57) and a constant wave amplitude a/D_m=0.152, where D_m is the mean diameter of the cylinder, is investigated using three dimensional (3D) large eddy simulation (LES) at a subcritical Reynolds number Re=3×10³, based on incoming free-stream velocity (U_∞) and D_m. Attention is paid to assimilating the effects of λ/D_m on the cylinder wake, including vortex shedding frequency, spanwise vortex formation length, streamwise velocity distribution, flow separation angle, 3D vortex structure, and turbulent kinetic energy (TKE) distribution. Based on the predominant role of λ/D_m in the near wake modification, three regimes are identified, i.e., regime I at λ/D_m<6.0, regime II at λ/D_m≈6.0 and regime III at λ/D_m>6.0. A dramatic decrease in fluid forces is observed at λ/D_m=6.06, about 16% and 93% reduction in time-averaged drag and fluctuating lift, respectively, compared to those of a smooth cylinder. We identified, for the first time, an optimum λ/D_m (=6.06) for the wavy cylinder with relatively large λ/D_m (>3.5) in the subcritical flow regime. The underlying mechanisms of force reduction are discussed, including the flow characteristics at the three λ/D_m regimes. A comparison is also made between the results of λ/D_m effects on the near wakes of a circular and a square cylinder.     

Underlying principle of efficient propulsion in flexible plunging foils

Passive flexibility was found to enhance propulsive efficiency in swimming animals.In this study,we numerically investigate the roles of structural resonance and hydrodynamic wake resonance in optimizing efficiency of a flexible plunging foil.The results indicates that（1）optimal efficiency is not necessarily achieved when the driving frequency matches the structural eigenfrequency;（2）optimal efficiency always occurs when the driving frequency matches the wake resonant frequency of the time averaged velocity profile.Thus,the underlying principle of efficient propulsion in flexible plunging foil is the hydrodynamic wake resonance,rather than the structural resonance.In addition,we also found that whether the efficiency can be optimized at the structural resonant point depends on the strength of the leading edge vortex relative to that of the trailing edge vortex.The result of this work provides new insights into the role of passive flexibility in flapping-based propulsion.     

The effect of a splitter plate on the flow around a finite prism

The effect of a wake-mounted splitter plate on the flow around a surface-mounted finite-height square prism was investigated experimentally in a low-speed wind tunnel. Measurements of the mean drag force and vortex shedding frequency were made at Re=7.4×10⁴ for square prisms of aspect ratios AR=9, 7, 5 and 3. Measurements of the mean wake velocity field were made with a seven-hole pressure probe at Re=3.7×10⁴ for square prisms of AR=9 and 5. The relative thickness of the boundary layer on the ground plane was δ/D=1.5–1.6 (where D is the side length of the prism). The splitter plates were mounted vertically from the ground plane on the wake centreline, with a negligible gap between the leading edge of the plate and rear of the prism. The splitter plate heights were always the same as the heights of prisms, while the splitter plate lengths ranged from L/D=1 to 7. Compared to previously published results for an “infinite” square prism, a splitter plate is less effective at drag reduction, but more effective at vortex shedding suppression, when used with a finite-height square prism. Significant reduction in drag was realized only for short prisms (of AR≤5) when long splitter plates (of L/D≥5) were used. In contrast, a splitter plate of length L/D=3 was sufficient to suppress vortex shedding for all aspect ratios tested. Compared to previous results for finite-height circular cylinders, finite-height square prisms typically need longer splitter plates for vortex shedding suppression. The effect of the splitter plate on the mean wake was to narrow the wake width close to the ground plane, stretch and weaken the streamwise vortex structures, and increase the lateral entrainment of ambient fluid towards the wake centreline. The splitter plate has little effect on the mean downwash. Long splitter plates resulted in the formation of additional streamwise vortex structures in the upper part of the wake.     

FLUID–STRUCTURE RESONANCE PRODUCED BY ONCOMING ALTERNATING VORTICES

The present investigation examines a simple fluid–structure interaction problem, which is represented by the unsteady response of an airfoil/blade to a Karman vortex street in an inviscid uniform flow. Two different cases were examined; one with a rigid airfoil/blade, where the structural stiffness is infinite, another with an elastic blade. In both cases, the flow remains attached to the airfoil/blade surface. A time-marching technique solving the Euler equations and a two-degree-of-freedom structural dynamic model is used to examine the interactions between the fluid and the structure. The interactions between the convected vortices and the structure modify the shed wake whose energy, in turn, feeds into the forces and moments acting on the structure. For a rigid airfoil/blade, it is found that the amplitude of the aerodynamic response is not proportional to the density of the oncoming vortex street, but depends on c/d , the ratio of the chord length (c) to the axial spacing (d) of the convected vortices. When the number of vortices per unit length is increased, the amplitudes of the aerodynamic response increase and then decrease even though the density of the vorticity keeps increasing and so is the energy of the excitation wake. Maxima are observed at c/d=0·5, 1·5 and 2·5. This behaviour is analogous to the structural resonance phenomenon and is labeled “aerodynamic resonance”. The existence of such an “aerodynamic resonance” is important to turbomachinery applications where the blade is elastic, the flow is unsteady and the shed vortices from the previous row are convected downstream by the mean flow. Thus, “aerodynamic resonance” alone or in conjunction with structural resonance could impact negatively on the fatigue life of turbine blades and their combined effects should be accounted for in blade design. A preliminary attempt to assess this impact has been carried out. It is found that the relative fatigue life of a blade could be reduced by four orders of magnitude as a result.     

Experimental investigation of a confined flow downstream of a circular cylinder centred between two parallel walls

In this work, we present an experimental study of the wall confinement effect on the wake formation behind a circular cylinder of diameter d_c=10 mm and of length L_c=30d_c. The experiments were performed in a water tunnel with the dimensions (length=300d_c, height=3d_c, span L_c=30d_c). The confinement rate was r=1/3 and the Reynolds number was in the range of 30–277. The experiments were done using 2-D PIV measurements. The first instability was delayed by the confinement and the von Kármán vortices characteristics are different from the unconfined case. Proper orthogonal decomposition (POD) of the flow was used for a filtering purpose and to extract the energetic contribution of the different modes. A low-order representation of the flow, constructed from the first pair of modes in the well-defined region of the flow, shows that von Kármán vortices are equivalent to vanishing progressive waves. Measurements done above the cylinder show the presence of 3-D span instabilities showing great similarities with “Mode A” and “Mode B” found in the unconfined case.     

Numerical study of large amplitude,nonsinusoidal motion and camber effects on pitching airfoil propulsion

The effects of large amplitude and nonsinusoidal motion on pitching airfoil aerodynamics for thrust generation were numerically studied with a 2-D NACA0012 airfoil used, and various 2-D NACA asymmetric airfoils were applied for camber effect study. The large amplitude effect study has been undertaken over a wide range of reduced frequency k (from 6 to 18) and pitching amplitude θ (from 5° to 30°) at Re=1.35×10⁴ with sinusoidal pitching profile used. It is shown that the large pitching amplitude results in much more thrust generated than that at low pitching amplitude and the increase of thrust with amplitude becomes slow when the amplitude reaches some degree. However, the propulsive efficiency noticeably decreases with the increase of θ at a fixed k.An adjustable parameter K was employed to realize various nonsinusoidal motions and the effect of nonsinusoidal motion was investigated with various unsteady parameters (θ, k) applied. The results reveal that nonsinusoidal motion has a noticeable effect on the aerodynamic performance, as it affects the instantaneous force coefficients, maximum thrust coefficients and flow structures. An increase in K results in a better thrust generation performance at fixed θ and k, especially for K>0. It is also shown that the larger K noticeably influences the wake pattern and induces a stronger reverse von Karman vortex street in the wake, which in turn leads to the increased thrust. The camber study was performed on various 2-D NACA airfoils with different cambers and camber locations undergoing sinusoidal pitching motion at θ=5° and Re=1.35×10⁴. It is found that varying camber offers little improvement in thrust generation performance.     

How a flexible tail improves the power extraction efficiency of a semi-activated flapping foil system: A numerical study

The improvement of power extraction of a semi-activated flapping foil system via the use of a flexible tail is numerically investigated in this work. A NACA0015 airfoil arranged in a two-dimensional laminar flow synchronously executes a forced pitching motion and an induced plunging motion. A flat plate attached to the trailing edge of the foil is utilized to model a tail, and thereby they are considered as a unit for the purpose of power extraction. The tail is either rigid or deformable. At a Reynolds number of 1100 and the position of the pitching axis at third chord, the effects of the mass and flexibility of the tail as well as the frequency of pitching motion on the net power extraction are systematically examined. It is found that compared to the foil with a rigid tail, the efficiency of net power extraction for the foil with a deformable tail can be improved. Based on the numerical analysis, it is indicated that the enhanced power extraction, which is caused by the increased lift force, directly contributes to the net efficiency improvement. Moreover, owing to high enhancement of power extraction, a flexible tail with high flexibility is recommended in the semi-activated flapping foil based power extraction system.     

Experimental benchmark of a free plunging wing with imposed flap oscillations

The validation of fluid–structure interaction solvers is difficult since there is a lack of experimental data. Therefore, in this work an aeroelastic experiment is presented. The focus is on the temporal coupling between fluid and structure dynamics. Issues in the spatial coupling are eliminated by using a rigid wing. The wing, with a harmonically actuated 0.2c trailing edge flap, has a degree of freedom in the plunge (vertical) direction. The wing has a chord of 0.5 m and is suspended with springs. The wing motion is constrained by a vertical rail system.For simplicity attached flow is desired and therefore the set angle of attack is α=0°. The Reynolds number is approximately Re=700 000 and the flap deflects over a range of about ±2°. The damped natural frequency of the structure expressed as a reduced frequency is about k=0.194 and measurements are performed for reduced flap frequencies ranging from k=0.1 to k=0.3. Displacements and time dependent aerodynamic forces are measured and for k=0.198 2-D PIV measurements are performed. The planar PIV measurements are used to intrinsically determine the unsteady loads using Noca׳s method.As expected the aeroelastic problem shows similarities with a viscously damped mass–damper–spring, meaning the maximum excursion of the wing is found near the system eigenfrequency. The lift is dominated by the flap motion and the effective angle of attack due to the motion introduces phase shifts of the lift signal with respect to the flap phase angle.The experiment has been set up and executed with the necessary precision, but small ambiguities are found in the lift and drag disqualifying the data for validation. Nevertheless the data set provides a clear insight into typical loads and motions and can be used for comparative studies. It can also be used to (re)design future experiments to improve the quality of the data to the desired level of accuracy for validation.     

The effect of a wake-mounted splitter plate on the flow around a surface-mounted finite-height circular cylinder

The influence of a wake-mounted splitter plate on the flow around a surface-mounted circular cylinder of finite height was investigated experimentally using a low-speed wind tunnel. The experiments were conducted at a Reynolds number of Re=7.4×10⁴ for cylinder aspect ratios of AR=9, 7, 5 and 3. The thickness of the boundary layer on the ground plane relative to the cylinder diameter was δ/D=1.5. The splitter plates were mounted on the wake centreline with negligible gap between the base of the cylinder and the leading edge of the plate. The lengths of the splitter plates, relative to the cylinder diameter, ranged from L/D=1 to 7, and the plate height was always equal to the cylinder height. Measurements of the mean drag force coefficient were obtained with a force balance, and measurements of the vortex shedding frequency were obtained with a single-component hot-wire probe situated in the wake of the cylinder–plate combination. Compared to the well-studied case involving an infinite circular cylinder, the splitter plate was found to be a less effective drag-reduction device for finite circular cylinders. Significant reduction in the mean drag coefficient was realized only for the finite circular cylinder of AR=9 with intermediate-length splitter plates of L/D=1–3. The mean drag coefficients of the other cylinders were almost unchanged. In terms of its effect on vortex shedding, a splitter plate of sufficient length was able to suppress Kármán vortex shedding for all of the finite circular cylinders tested. For AR=9, vortex shedding suppression occurred for L/D≥5, which is similar to the case of the infinite circular cylinder. For the smaller-aspect-ratio cylinders, however, the splitter plate was more effective than what occurs for the infinite circular cylinder: for AR=3, vortex shedding suppression occurred for all of the splitter plates tested (L/D≥1); for AR=5 and 7, vortex shedding suppression occurred for L/D≥1.5.