Correction to 'On the possibility (or lack thereof) of agreement between experiment and computation of flows over wings at moderate Reynolds number'.

[This corrects the article DOI: 10.1098/rsfs.2016.0076.].

On the possibility (or lack thereof) of agreement between experiment and computation of flows over wings at moderate Reynolds number.

The flight of many birds and bats, and their robotic counterparts, occurs over a range of chord-based Reynolds numbers from 1 × 104 to 1.5 × 105. It is precisely over this range where the aerodynamics of simple, rigid, fixed wings becomes extraordinarily sensitive to small changes in geometry and the environment, with two sets of consequences. The first is that practical lifting devices at this scale will likely not be simple, rigid, fixed wings. The second is that it becomes non-trivial to make baseline comparisons for experiment and computation, when either one can be wrong. Here we examine one ostensibly simple case of the NACA 0012 aerofoil and make careful comparison between the technical literature, and new experiments and computations. The agreement (or lack thereof) will establish one or more baseline results and some sensitivities around them. The idea is that the diagnostic procedures will help to guide comparisons and predictions in subsequent more complex cases.

Leading-edge vortex improves lift in slow-flying bats.

Staying aloft when hovering and flying slowly is demanding. According to quasi-steady-state aerodynamic theory, slow-flying vertebrates should not be able to generate enough lift to remain aloft. Therefore, unsteady aerodynamic mechanisms to enhance lift production have been proposed. Using digital particle image velocimetry, we showed that a small nectar-feeding bat is able to increase lift by as much as 40% using attached leading-edge vortices (LEVs) during slow forward flight, resulting in a maximum lift coefficient of 4.8. The airflow passing over the LEV reattaches behind the LEV smoothly to the wing, despite the exceptionally large local angles of attack and wing camber. Our results show that the use of unsteady aerodynamic mechanisms in flapping flight is not limited to insects but is also used by larger and heavier animals.

Vortex wake and flight kinematics of a swift in cruising flight in a wind tunnel.

In this paper we describe the flight characteristics of a swift (Apus apus) in cruising flight at three different flight speeds (8.0, 8.4 and 9.2 m s(-1)) in a low turbulence wind tunnel. The wingbeat kinematics were recorded by high-speed filming and the wake of the bird was visualized by digital particle image velocimetry (DPIV). Certain flight characteristics of the swift differ from those of previously studied species. As the flight speed increases, the angular velocity of the wingbeat remains constant, and so as the wingbeat amplitude increases, the frequency decreases accordingly, as though the flight muscles were contracting at a fixed rate. The wings are also comparatively inflexible and are flexed or retracted rather little during the upstroke. The upstroke is always aerodynamically active and this is reflected in the wake, where shedding of spanwise vorticity occurs throughout the wingbeat. Although the wake superficially resembles those of other birds in cruising flight, with a pair of trailing wingtip vortices connected by spanwise vortices, the continuous shedding of first positive vorticity during the downstroke and then negative vorticity during the upstroke suggests a wing whose circulation is gradually increasing and then decreasing during the wingbeat cycle. The wake (and implied wing aerodynamics) are not well described by discrete vortex loop models, but a new wake-based model, where incremental spanwise and streamwise variations of the wake impulse are integrated over the wingbeat, shows good agreement of the vertical momentum flux with the required weight support. The total drag was also estimated from the wake alone, and the calculated lift:drag ratio of approximately 13 for flapping flight is the highest measured yet for birds.

The implications of low-speed fixed-wing aerofoil measurements on the analysis and performance of flapping bird wings.

Bird flight occurs over a range of Reynolds numbers (Re; 10(4) < or = Re < or = 10(5), where Re is a measure of the relative importance of inertia and viscosity) that includes regimes where standard aerofoil performance is difficult to predict, compute or measure, with large performance jumps in response to small changes in geometry or environmental conditions. A comparison of measurements of fixed wing performance as a function of Re, combined with quantitative flow visualisation techniques, shows that, surprisingly, wakes of flapping bird wings at moderate flight speeds admit to certain simplifications where their basic properties can be understood through quasi-steady analysis. Indeed, a commonly cited measure of the relative flapping frequency, or wake unsteadiness, the Strouhal number, is seen to be approximately constant in accordance with a simple requirement for maintaining a moderate local angle of attack on the wing. Together, the measurements imply a fine control of boundary layer separation on the wings, with implications for control strategies and wing shape selection by natural and artificial fliers.

Bat flight generates complex aerodynamic tracks.

The flapping flight of animals generates an aerodynamic footprint as a time-varying vortex wake in which the rate of momentum change represents the aerodynamic force. We showed that the wakes of a small bat species differ from those of birds in some important respects. In our bats, each wing generated its own vortex loop. Also, at moderate and high flight speeds, the circulation on the outer (hand) wing and the arm wing differed in sign during the upstroke, resulting in negative lift on the hand wing and positive lift on the arm wing. Our interpretations of the unsteady aerodynamic performance and function of membranous-winged, flapping flight should change modeling strategies for the study of equivalent natural and engineered flying devices.

Wake structure and wingbeat kinematics of a house-martin Delichon urbica.

The wingbeat kinematics and wake structure of a trained house martin in free, steady flight in a wind tunnel have been studied over a range of flight speeds, and compared and contrasted with similar measurements for a thrush nightingale and a pair of robins. The house martin has a higher aspect ratio (more slender) wing, and is a more obviously agile and aerobatic flyer, catching insects on the wing. The wingbeat is notable for the presence at higher flight speeds of a characteristic pause in the upstroke. The essential characteristics of the wing motions can be reconstructed with a simple two-frequency model derived from Fourier analysis. At slow speeds, the distribution of wake vorticity is more simple than for the other previously measured birds, and the upstroke does not contribute to weight support. The upstroke becomes gradually more significant as the flight speed increases, and although the vortex wake shows a signature of the pause phase, the global circulation measurements are otherwise in good agreement with surprisingly simple aerodynamic models, and with predictions across the different species, implying quite similar aerodynamic performance of the wing sections. The local Reynolds numbers of the wing sections are sufficiently low that the well-known instabilities of attached laminar flows over lifting surfaces, which are known to occur at two to three times this value, may not develop.

Vortex wakes generated by robins Erithacus rubecula during free flight in a wind tunnel.

The wakes of two individual robins were measured in digital particle image velocimetry (DPIV) experiments conducted in the Lund wind tunnel. Wake measurements were compared with each other, and with previous studies in the same facility. There was no significant individual variation in any of the measured quantities. Qualitatively, the wake structure and its gradual variation with flight speed were exactly as previously measured for the thrush nightingale. A procedure that accounts for the disparate sources of circulation spread over the complex wake structure nevertheless can account for the vertical momentum flux required to support the weight, and an example calculation is given for estimating drag from the components of horizontal momentum flux (whose net value is zero). The measured circulations of the largest structures in the wake can be predicted quite well by simple models, and expressions are given to predict these and other measurable quantities in future bird flight experiments.

The relationship between wingbeat kinematics and vortex wake of a thrush nightingale.

The wingbeat kinematics of a thrush nightingale Luscinia luscinia were measured for steady flight in a wind tunnel over a range of flight speeds (5-10 m s(-1)), and the results are interpreted and discussed in the context of a detailed, previously published, wake analysis of the same bird. Neither the wingbeat frequency nor wingbeat amplitude change significantly over the investigated speed range and consequently dimensionless measures that compare timescales of flapping vs. timescales due to the mean flow vary in direct proportion to the mean flow itself, with no constant or slowly varying intervals. The only significant kinematic variations come from changes in the upstroke timing (downstroke fraction) and the upstroke wing folding (span ratio), consistent with the gradual variations, primarily in the upstroke wake, previously reported. The relationship between measured wake geometry and wingbeat kinematics can be qualitatively explained by presumed self-induced convection and deformation of the wake between its initial formation and later measurement, and varies in a predictable way with flight speed. Although coarse details of the wake geometry accord well with the kinematic measurements, there is no simple explanation based on these observed kinematics alone that accounts for the measured asymmetries of circulation magnitude in starting and stopping vortex structures. More complex interactions between the wake and wings and/or body are implied.

Comparing fluid mechanics models with experimental data.

The art of modelling the physical world lies in the appropriate simplification and abstraction of the complete problem. In fluid mechanics, the Navier-Stokes equations provide a model that is valid under most circumstances germane to animal locomotion, but the complexity of solutions provides strong incentive for the development of further, more simplified practical models. When the flow organizes itself so that all shearing motions are collected into localized patches, then various mathematical vortex models have been very successful in predicting and furthering the physical understanding of many flows, particularly in aerodynamics. Experimental models have the significant added convenience that the fluid mechanics can be generated by a real fluid, not a model, provided the appropriate dimensionless groups have similar values. Then, analogous problems can be encountered in making intelligible but independent descriptions of the experimental results. Finally, model predictions and experimental results may be compared if, and only if, numerical estimates of the likely variations in the tested quantities are provided. Examples from recent experimental measurements of wakes behind a fixed wing and behind a bird in free flight are used to illustrate these principles.

A family of vortex wakes generated by a thrush nightingale in free flight in a wind tunnel over its entire natural range of flight speeds.

In view of the complexity of the wing-beat kinematics and geometry, an important class of theoretical models for analysis and prediction of bird flight performance entirely, or almost entirely, ignores the action of the wing itself and considers only the resulting motions in the air behind the bird. These motions can also be complicated, but some success has previously been recorded in detecting and measuring relatively simple wake structures that can sometimes account for required quantities used to estimate aerodynamic power consumption. To date, all bird wakes, measured or presumed, seem to fall into one of two classes: the closed-loop, discrete vortex model at low flight speeds, and the constant-circulation, continuous vortex model at moderate to high speeds. Here, novel and accurate quantitative measurements of velocity fields in vertical planes aligned with the freestream are used to investigate the wake structure of a thrush nightingale over its entire range of natural flight speeds. At most flight speeds, the wake cannot be categorised as one of the two standard types, but has an intermediate structure, with approximations to the closed-loop and constant-circulation models at the extremes. A careful accounting for all vortical structures revealed with the high-resolution technique permits resolution of the previously unexplained wake momentum paradox. All the measured wake structures have sufficient momentum to provide weight support over the wingbeat. A simple model is formulated and explained that mimics the correct, measured balance of forces in the downstroke- and upstroke-generated wake over the entire range of flight speeds. Pending further work on different bird species, this might form the basis for a generalisable flight model.

Uncertainty calculations for theoretical flight power curves.

The comparison of theoretical and experimental estimates of the mechanical power requirement for flight is currently impossible owing to the absence of complete experimental data based on mechanical power, as opposed to measurements of metabolic rates. Nevertheless, comparisons of measured and predicted characteristic speeds, and inferred power curves are frequently made, despite the total absence of uncertainty estimates of the theoretically predicted quantities. Here the method for correct calculation of uncertainty estimates in mechanical power models is outlined in detail, and analytical and numerical results are derived for realistic examples. The sensitivity of the calculated variations in power requirement varies greatly among the independent variables, and the practical and theoretical consequences of this variation are discussed. Pending the arrival of appropriate experimental measurements, it is now possible, in principle, to make quantitative comparisons with theoretical predictions.