An experimental study of the effects of pitch-pivot-point location on the propulsion performance of a pitching airfoil

An experimental investigation was conducted to characterize the evolution of the unsteady vortex structures in the wake of a pitching airfoil with the pitch-pivot-point moving from 0.16C to 0.52C (C is the chord length of the airfoil). The experimental study was conducted in a low-speed wind tunnel with a symmetric NACA0012 airfoil model in pitching motion under different pitching kinematics (i.e., reduced frequency k=3.8–13.2). A high-resolution particle image velocimetry (PIV) system was used to conduct detailed flow field measurements to quantify the characteristics of the wake flow and the resultant propulsion performance of the pitching airfoil. Besides conducting “free-run” PIV measurements to determine the ensemble-averaged velocity distributions in the wake flow, “phase-locked” PIV measurements were also performed to elucidate further details about the behavior of the unsteady vortex structures. Both the vorticity–moment theorem and the integral momentum theorem were used to evaluate the effects of the pitch-pivot-point location on the propulsion performance of the pitching airfoil. It was found that the pitch-pivot-point would affect the evolution of the unsteady wake vortices and resultant propulsion performance of the pitching airfoil greatly. Moving the pitch-pivot-point of the pitching airfoil can be considered as adding a plunging motion to the original pitching motion. With the pitch-pivot-point moving forward (or backward), the added plunging motion would make the airfoil trailing edge moving in the same (or opposite) direction as of the original pitching motion, which resulted in the generated wake vortices and resultant thrust enhanced (or weakened) by the added plunging motion.     

Wake visualization of a heaving and pitching foil in a soap film

Many fish depend primarily on their tail beat for propulsion. Such a tail is commonly modeled as a two-dimensional flapping foil. Here we demonstrate a novel experimental setup of such a foil that heaves and pitches in a soap film. The vortical flow field generated by the foil correlates with thickness variations in the soap film, which appear as interference fringes when the film is illuminated with a monochromatic light source (we used a high-frequency SOX lamp). These interference fringes are subsequently captured with high-speed video (500 Hz) and this allows us to study the unsteady vortical field of a flapping foil. The main advantage of our approach is that the flow fields are time and space resolved and can be obtained time-efficiently. The foil is driven by a flapping mechanism that is optimized for studying both fish swimming and insect flight inside and outside the behavioral envelope. The mechanism generates sinusoidal heave and pitch kinematics, pre-described by the non-dimensional heave amplitude (0–6), the pitch amplitude (0°–90°), the phase difference between pitch and heave (0°–360°), and the dimensionless wavelength of the foil (3–18). We obtained this wide range of wavelengths for a foil 4 mm long by minimizing the soap film speed (0.25 m s⁻¹) and maximizing the flapping frequency range (4–25 Hz). The Reynolds number of the foil is of order 1,000 throughout this range. The resulting setup enables an effective assessment of vortex wake topology as a function of flapping kinematics. The efficiency of the method is further improved by carefully eliminating background noise in the visualization (e.g., reflections of the mechanism). This is done by placing mirrors at an angle behind the translucent film such that the camera views the much more distant and out-of-focus reflections of the black laboratory wall. The resulting high-quality flow visualizations require minimal image processing for flow interpretation. Finally, we demonstrate the effectiveness of our setup by visualizing the vortex dynamics of the flapping foil as a function of pitch amplitude by assessing the symmetry of the vortical wake.     

Numerical investigations of the vortex interactions for a flow over a pitching foil at different stages

The fluid–structure interaction is investigated numerically for a two-dimensional flow (Re=2.5·10⁶) over a sinusoid-pitching foil by the SST (Shear Stress Transport) k–ω model. Although discrepancies in the downstroke phase, which are also documented in other numerical studies, are observed by comparing with experimental results, our current numerical results are sufficient to predict the mean features and qualitative tendencies of the dynamic stall phenomenon. These discrepancies are evaluated carefully from the numerical and experimental viewpoints.In this study, we have utilized Λ, which is the normalized second invariant of the velocity gradient tensor, to present the evolution of the Leading Edge Vortex (LEV) and Trailing Edge Vortex (TEV). The convective, pressure, and diffusion terms during the dynamic stall process are discussed based on the transport equation of Λ. It is found that the pressure term dominates the rate of the change of the rotation strength inside the LEV. This trend can hardly be observed directly by using the vorticity transport equation due to the zero baroclinic term for the incompressible flow.The mechanisms to delay the stall are categorized based on the formation of the LEV. At the first stage before the formation of the LEV in the upper surface, the pitching foil provides extra momentum into the fluid flows to resist the flow separation, and hence the stall is delayed. At the second stage, a low-pressure area travels with the evolution of the LEV such that the lift still can be maintained. Three short periods at the second stage corresponds to different flow patterns during the dynamic stall, and these short periods can be distinguished according to the trend of the pressure variation inside the LEV. The lift stall occurs when a reverse flow from the lower surface is triggered during the shedding of the LEV. For a reduced frequency k_f=0.15, the formation of the TEV happens right after the lift stall, and the lift can drop dramatically. With a faster reduced frequency k_f=0.25, the shedding of the LEV is postponed into the downstroke, and the interaction between the LEV and TEV becomes weaker correspondingly. Thus, the lift drops more gently after the stall. In order to acquire more reliable numerical results within the downstroke phase, the Large Eddy Simulation (LES), which is capable of better predictions for the laminar-to-turbulent transition and flow reattachment process, will be considered as the future work.     

Experimental assessment of performance characteristics for pitching flexible propulsors

Using a flexible hydrodynamic foil that pitches to produce thrust, the most pertinent aspects of a fish-like propulsion system are replicated in a controlled environment. The pitching and flexing combination creates a hydroelastic coupling in which the fluid and flexible foil simultaneously affect each other's behavior. The project investigated relationships for the propulsors’ thrust and efficiency performance to gain a better understanding of the dynamic interaction with the surrounding fluid. The analysis was conducted through reduction of the measured force and torque data. The experiments took place in a large recirculating water channel, using full span flexible propulsor models and a higher Reynolds number than previous flexible propulsor experiments. The propulsor pitched about a fixed axis at its quarter chord, with a six-axis load cell measuring the forces and torques on the shaft. The efficiency of the propulsor and the Coefficients of Thrust and Lift are presented as functions of both Strouhal Number and Stiffness Coefficient. The ensemble data will facilitate the engineering of fish-like propulsion systems for future application of this technology.     

Effect of uncertainty on the bifurcation behavior of pitching airfoil stall flutter

In this paper the effect of system parametric uncertainty on the stall flutter bifurcation behavior of a pitching airfoil is studied. The aerodynamic moment on the two-dimensional rigid airfoil with nonlinear torsional stiffness is computed using the ONERA dynamic stall model. The pitch natural frequency, a cubic structural nonlinearity parameter, and the structural equilibrium angle are assumed to be uncertain. The effect on the amplitude of the response, the bifurcation of the probability distribution, and the flutter boundary is considered. It is demonstrated that the system parametric uncertainty results already in 5% probability of pitching stall flutter at a 12.5% earlier position than the point where a deterministic analysis would predict unstable behavior. Probabilistic collocation is found to be more efficient than the Galerkin polynomial chaos method and Monte Carlo simulation for modeling uncertainty in the post-bifurcation domain.     

Unsteady response of airfoils due to small-scale pitching motion with considerations for foil thickness and wake motion

Unsteady pressures, forces, and pitching moments generated by foils experiencing vibratory motion in an incompressible, attached flow configuration are studied within this work. Specifically, two-dimensional, unsteady potential flow and unsteady Reynolds-Averaged Navier–Stokes calculations are performed on various Joukowski foils undergoing sinusoidal, variable amplitude, small-scale pitching motion at a chord-based Reynolds number of 10${}^6$ over a range of reduced frequencies between 0.01–100. These calculated results from both approaches are compared directly to predictions from implementing the Theodorsen model, which treats foils as infinitely thin, flat plates that shed a planar sheet of vorticity. The effects of relaxing these seemingly strict conditions are explored, and the particular terms which control the unsteady responses are identified and discussed. For increasing pitch amplitudes and reduced frequencies the shed wake is seen to become quite non-planar and to form coherent vortex structures. Despite this wake behavior, the normalized airfoil responses at the disturbance reduced frequency are seen to be largely unaffected. However, non-negligible responses are generated across a wide range of other frequencies. Potential flow calculations for symmetric Joukowski foils show that there is marginal effect of foil thickness at reduced frequencies less than one. For higher reduced frequency conditions however, the unsteady lift response is seen to experience both an amplification of level and a phase shift relative to the Theodorsen model. A specific augmenting expression is developed for this behavior through analysis within the potential flow framework.     

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.     

Influence of heat transfer on the aerodynamic performance of a plunging and pitching NACA0012 airfoil at low Reynolds numbers

The unsteady low Reynolds number aerodynamics phenomena around flapping wings are addressed in several investigations. Elsewhere, airfoils at higher Mach numbers and Reynolds numbers have been treated quite comprehensively in the literature. It is duly noted that the influence of heat transfer phenomena on the aerodynamic performance of flapping wings configurations is not well studied. The objective of the present study is to investigate the effect of heat transfer upon the aerodynamic performance of a pitching and plunging NACA0012 airfoil in the low Reynolds number flow regime with particular emphasis upon the airfoil's lift and drag coefficients. The compressible Navier–Stokes equations are solved using a finite volume method. To consider the variation of fluid properties with temperature, the values of dynamic viscosity and thermal diffusivity are evaluated with Sutherland's formula and the Eucken model, respectively. Instantaneous and mean lift and drag coefficients are calculated for several temperature differences between the airfoil surface and freestream within the range 0–100 K. Simulations are performed for a prescribed airfoil motion schedule and flow parameters. It is learnt that the aerodynamic performance in terms of the lift C_L and drag C_D behavior is strongly dependent upon the heat transfer rate from the airfoil to the flow field. In the plunging case, the mean value of C_D tends to increase, whereas the amplitude of C_L tends to decrease with increasing temperature difference. In the pitching case, on the other hand, the mean value and the amplitude of both C_D and C_L decrease. A spectral analysis of C_D and C_L in the pitching case shows that the amplitudes of both C_D and C_L decrease with increasing surface temperature, whereas the harmonic frequencies are not affected.     

Hydrodynamic problem of the pitching of a catamaran on shallow water

The problem of calculating the pitching of multihull vessels, which are widely used in world shipping, has been inadequately developed. Some investigations [1, 2] are far from practical realization, while others [3] are based on too crude simplifying assumptions. In the present paper, the method of [4] is used to solve the hydrodynamic problem of the pitching of a catamaran. Expressions are obtained for the potential of the perturbed velocities of the liquid, the wave damping, and the perturbing forces, and the results are given of a calculation of the hydrodynamic characteristics.Translated from Izvestlya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 147–151, March–April, 1980.     

Numerical analysis on the propulsive performance and vortex shedding of fish‐like travelling wavy plate

Numerical analysis is carried out to investigate viscous flow over a travelling wavy plate undergoing lateral motion in the form of a streamwise travelling wave, which is similar to the backbone undulation of swimming fish. The two‐dimensional incompressible Navier–Stokes equations are solved using the finite element technique with the deforming‐spatial‐domain/stabilized space–time formulation. The objective of this study is to elucidate hydrodynamic features of flow structure and vortex shedding near the travelling wavy plate and to get into physical insights to the understanding of fish‐like swimming mechanisms in terms of drag reduction and optimal propulsive performance. The effects of some typical parameters, including the phase speed, amplitude, and relative wavelength of travelling wavy plate, on the flow structures, the forces, and the power consumption required for the propulsive motion of the plate are analysed. These results predicted by the present numerical analysis are well consistent with the available data obtained for the wave‐like swimming motion of live fish in nature. Copyright © 2005 John Wiley & Sons, Ltd.     

On the dynamics of a spinning top under high-frequency excitation. Part II: pivot point under horizontal harmonic vibration

We investigate the dynamics of a spinning top whose pivot point undergoes a small-amplitude high-frequency horizontal vibration. The method of direct partition of motion is used to obtain an autonomous two-degree-of-freedom system governing the leading-order slow dynamics of the top’s nutation and precession angles. We show that the fast vibration leads to loss of stability of the upright “sleeping top” equilibrium state of the spinning top. We also show the existence of two new apparent equilibrium states that correspond to special solutions in which the spinning top is locked at constant nutation with the precession angle being aligned with the direction of excitation. We refer to those as “skewed sleeping top” states. We derive the condition for which these states exist and show that they are stable. The results are verified through numerical integration of the full non-autonomous system. In addition, we illustrate how these states allow the control of the attitude of the top through slow variation of the amplitude of fast excitation.     

An analogy between a gravity modulated porous layer heated from below and the inverted pendulum with an oscillating pivot point

The governing equations for the inverted pendulum is developed and shown to be analogous to the gravity modulated porous layer heated from below. In particular the temperature in a gravity modulated porous layer heated from below (R>0-unstable) is likened to the motion of an inverted pendulum (unstable). As gravity modulation stabilizes convection, Transport Porous Med. 57 (2004), 113, it is found that oscillating the pivot point of an unstable, inverted pendulum stabilizes the motion.     

Lagrangian-based numerical investigation of aerodynamic performance of an oscillating foil

The dynamic stall problem for blades is related to the general performance of wind turbines, where a varying flow field is introduced with a rapid change of the effective angle of attack (AOA). The objective of this work is to study the aerodynamic performance of a sinusoidally oscillating NACA0012 airfoil. The coupled \(k{-}\omega \) Menter’s shear stress transport (SST) turbulence model and \(\gamma {-}Re_{\uptheta }\) transition model were used for turbulence closure. Lagrangian coherent structures (LCS) were utilized to analyze the dynamic behavior of the flow structures. The computational results were supported by the experiments. The results indicated that this numerical method can well describe the dynamic stall process. For the case with reduced frequency \(K = 0.1\), the lift and drag coefficients increase constantly with increasing angle prior to dynamic stall. When the AOA reaches the stall angle, the lift and drag coefficients decline suddenly due to the interplay between the first leading- and trailing-edge vortex. With further increase of the AOA, both the lift and drag coefficients experience a secondary rise and fall process because of formation and shedding of the secondary vortex. The results also reveal that the dynamic behavior of the flow structures can be effectively identified using the finite-time Lyapunov exponent (FTLE) field. The influence of the reduced frequency on the flow structures and energy extraction efficiency in the dynamic stall process is further discussed. When the reduced frequency increases, the dynamic stall is delayed and the total energy extraction efficiency is enhanced. With \(K = 0.05\), the amplitude of the dynamic coefficients fluctuates more significantly in the poststall process than in the case of \(K = 0.1\).     

Numerical experiments on the transient motions of a flapping foil

The flow field generated by a foil during transient motions is investigated by means of numerical experiments. The numerical simulations have some advantages with respect to laboratory experiments. Indeed, having access to the velocity and pressure fields both in space and in time, it is possible to 'measure' quantities like vorticity, forces and torques which are quite difficult to obtain in laboratory. Moreover, data can be easily gained for different foil kinematics. The obtained results show that the time history of the propulsive force strongly depends on the details of the kinematics of the foil. Moreover, the numerical simulations have allowed to understand the main mechanisms employed by fish to propel themselves during fast starts and to identify the values of the parameters providing optimal propulsive performances.     

Forced precession in a spinning wheel supported on a rotating pivot

This paper deals with the mechanics involved in a spinning wheel of which the pivot is not fixed as usual but is forced to rotate along the circumference of a circle on the horizontal plane. The usual Euler equations are extended so that, in addition to the three well-known rotations (Euler angles), they also include a fourth one related to the rotation of the motor that induces the forced precession. This study aims at offering a first insight in one of the renowned Laithwaite's experiments. The derived theoretical expressions are accompanied by computer simulation.     

Numerical investigations into the asymmetric effects on the aerodynamic response of a pitching airfoil

The effects of asymmetric sinusoidal motion on pitching airfoil aerodynamics were studied by numerical simulations for 2-D flow around a NACA0012 airfoil at Re=1.35×10⁵. Various unsteady parameters (amplitude of oscillation, d; reduced frequency, k) were applied to investigate the effect of asymmetry parameter S on the instantaneous force coefficients and flow patterns. The results reveal that S has a noticeable effect on the aerodynamic performance, as it affects the instantaneous force coefficient, maximum lift and drag coefficient, hysteresis loops and the flow structures.     

A numerical study of the propulsive efficiency of a flapping hydrofoil

A computational fluid dynamics study of the swimming efficiency of a two‐dimensional flapping hydrofoil at a Reynolds number of 1100 is presented. The model accounts fully for viscous effects that are particularly important when flow separation occurs. The model uses an arbitrary Lagrangian–Eulerian (ALE) method to track the moving boundaries of oscillatory and flapping bodies. A parametric analysis is presented of the variables that affect the motion of the hydrofoil as it moves through the flow along with flow visualizations in an attempt to quantify and qualify the effect that these variables have on the performance of the hydrofoil. Copyright © 2003 John Wiley & Sons, Ltd.     

Computational study of a pitching bio-inspired corrugated airfoil

The phenomenon of insect flight has been of scientific interest for many years and is more recently inspiring modern engineering devices such as Micro Aerial Vehicles (MAVs). Insect flight is characterized by unsteady fluid dynamics at low Reynolds numbers. The importance of viscous effects to the successful flapping flight of insects has been identified and with the current state of computing and Computational Fluid Dynamics (CFD) these effects can now be studied in detail. The present work attempts to simplify this complex phenomenon by considering symmetric oscillating rotational motion of a wing (pitching). What is of interest in this study is how the shape of a corrugated idealized insect wing affects the performance and flow characteristics around the pitching wing. Two dimensional CFD on an oscillating wing has been performed and reported. Measurements were taken to ensure the accuracy of the computational solution and the results validated against experimental PIV results. A range of frequencies and rotational amplitudes have been investigated. Lift and drag coefficients have been analyzed for all cases to quantify the effects of unsteady flow features on the performance of the oscillating wing. It was found that the wing shape used in this study resulted in the viscous features formed on the top of the wing exhibiting high sensitivity to the oscillating conditions and these influenced the performance of the wing. The flow features formed on the bottom of the wing remained similar throughout the cases tested. In the pitching regime this wing profile did not perform as well as published results for smooth airfoils in terms of thrust and propulsion efficiency. However this may be due to reduced frequency effects becoming important at our high pitching amplitude which need to be investigated further. There may be other oscillatory regimes that more accurately represent flapping flight in which the corrugated foil outperforms a smooth counterpart but these are yet to be investigated. Further research in this area may help answer the question as to how evolutionarily significant other benefits of a corrugated wing, such as being light and strong, are compared to its aerodynamic properties, the present results seem to favor the former.