Lift coefficient

Abstract: A method is presented which enables the computation of the bed-load transport as the product of the saltation height, the particle velocity and the bed-load concentration. The equations of motions for a solitary particle are solved numerically to determine the saltation height and particle velocity. Experiments with gravel particles (transported as bed load) are selected to calibrate the mathematical model using the lift coefficient as a free parameter. The model is used to compute the saltation heights and lengths for a range of flow conditions. The computational results are used to determine simple relationships for the saltation characteristics. Measured transport rates of the bed load are used to compute the sediment concentration in the bed-load layer. A simple expression specifying the bed-load concentration as a function of the flow and sediment conditions is proposed. A verification analysis using about 600 (alternative) data shows that about 77% of the predicted bed-load-transport rates are within 0.5 and 2 times the observed values.

Abstract: 1. On the assumption that steady-state aerodynamics applies, simple analytical expressions are derived for the average lift coefficient, Reynolds number, the aerodynamic power, the moment of inertia of the wing mass and the dynamic efficiency in animals which perform normal hovering with horizontally beating wings. 2. The majority of hovering animals, including large lamellicorn beetles and sphingid moths, depend mainly on normal aerofoil action. However, in some groups with wing loading less than 10 N m -2 (1 kgf m -2 ), non-steady aerodynamics must play a major role, namely in very small insects at low Reynolds number, in true hover-flies (Syrphinae), in large dragonflies (Odonata) and in many butterflies (Lepidoptera Rhopalocera). 3. The specific aerodynamic power ranges between 1.3 and 4.7 WN -1 (11-40 cal h -1 gf -1 ) but power output does not vary systematically with size, inter alia because the lift/drag ratio deteriorates at low Reynolds number. 4. Comparisons between metabolic rate, aerodynamic power and dynamic efficiency show that the majority of insects require and depend upon an effective elastic system in the thorax which counteracts the bending moments caused by wing inertia. 5. The free flight of a very small chalcid wasp Encarsia formosa has been analysed by means of slow-motion films. At this low Reynolds number (10-20), the high lift co-efficient of 2 or 3 is not possible with steady-state aerodynamics and the wasp must depend almost entirely on non-steady flow patterns. 6. The wings of Encarsia are moved almost horizontally during hovering, the body being vertical, and there are three unusual phases in the wing stroke: the clap , the fling and the flip . In the clap the wings are brought together at the top of the morphological upstroke. In the fling, which is a pronation at the beginning of the morphological downstroke, the opposed wings are flung open like a book, hinging about their posterior margins. In the flip, which is a supination at the beginning of the morphological upstroke, the wings are rapidly twisted through about 180°. 7. The fling is a hitherto undescribed mechanism for creating lift and for setting up the appropriate circulation over the wing in anticipation of the downstroke. In the case of Encarsia the calculated and observed wing velocities at which lift equals body weight are in agreement, and lift is produced almost instantaneously from the beginning of the downstroke and without any Wagner effect. The fling mechanism seems to be involved in the normal flight of butterflies and possibly of Drosophila and other small insects. Dimensional and other considerations show that it could be a useful mechanism in birds and bats during take-off and in emergencies. 8. The flip is also believed to be a means of setting up an appropriate circulation around the wing, which has hitherto escaped attention; but its operation is less well understood. It is not confined to Encarsia but operates in other insects, not only at the beginning of the upstroke (supination) but also at the beginning of the downstroke where a flip (pronation) replaces the clap and fling of Encarsia . A study of freely flying hover-flies strongly indicates that the Syrphinae (and Odonata) depend almost entirely upon the flip mechanism when hovering. In the case of these insects a transient circulation is presumed to be set up before the translation of the wing through the air, by the rapid pronation (or supination) which affects the stiff anterior margin before the soft posterior portions of the wing. In the flip mechanism vortices of opposite sense must be shed, and a Wagner effect must be present. 9. In some hovering insects the wing twistings occur so rapidly that the speed of propagation of the elastic torsional wave from base to tip plays a significant role and appears to introduce beneficial effects. 10. Non-steady periods, particularly flip effects, are present in all flapping animals and they will modify and become superimposed upon the steady-state pattern as described by the mathematical model presented here. However, the accumulated evidence indicates that the majority of hovering animals conform reasonably well with that model. 11. Many new types of analysis are indicated in the text and are now open for future theoretical and experimental research.

Abstract: x(/3) Trailing vortices do not decay by simple diffusion. Usually they undergo a symmetric and nearly sinusoidal instability, until eventually they join at intervals to form a train of vortex rings. The present theory accounts for the instability during the early stages of its growth. The vortices are idealized as interacting lines; their core diameters are taken into account by a cutoff in the line integral representing self-induction. The equation relating induced velocity to vortex displacement gives rise to an eigenvalue problem for the growth rate of sinusoidal perturbations. Stability is found to depend on the products of vortex separation 6 and cutoff distance d times the perturbation wavenumber. Depending on those products, both symmetric and antisymmetric eigenmodes can be unstable, but only the symmetric mode involves strongly interacting long waves. An argument is presented that d/b = 0.063 for the vortices trailing from an elliptically loaded wing. In that case, the maximally unstable long wave has a length 8.66 and grows by a factor e in a time 9.4(^4#/CL)(6/F0), where AR is the aspect ratio, CL is the lift coefficient, and V0 is the speed of the aircraft. The vortex displacements are symmetric and are confined to fixed planes inclined at 48° to the horizontal.

Abstract: Apart from providing some new experimental data the paper reviews previous investigations concerning fluctuating lift acting on a stationary circular cylinder in cross-flow. In particular, effects of Reynolds number in the nominal case of an infinitely long and nonconfined cylinder in a smooth oncoming flow are discussed. The Reynolds number range covered is from about Re = 47 to 2 x 10(5), i.e., from the onset of vortex shedding up to the end of the subcritical regime. At the beginning of the subcritical regime (Reesimilar or equal to0.3 x 10(3)) a spanwise correlation length of about 30 cylinder diameters is indicated, the correlation function being based on near-cylinder velocity fluctuations in outer parts of the separated shear layer. In between Reynolds numbers 1.6 x 10(3) and 20 x 10(3), an approximate 10-fold increase in the sectional r.m.s. lift coefficient is indicated. This range contains a fundamental change-over from one flow state to another, starting off at Re similar or equal to 5 x 103 and seemingly fully developed at Re similar or equal to 8 x 10(3). (C) 2002 Elsevier Science Ltd. All rights reserved. (Less)

Abstract: Trajectories of single air bubbles in simple shear flows of glycerol–water solution were measured to evaluate transverse lift force acting on single bubbles. Experiments were conducted under the conditions of , 1.39⩽Eo⩽5.74 and , where M is the Morton number, Eo the Eotvos number and dVL/dy the velocity gradient of the shear flow. A net transverse lift coefficient CT was evaluated by making use of all the measured trajectories and an equation of bubble motion. It was confirmed that CT for small bubbles is a function of the bubble Reynolds number Re, whereas CT for larger bubbles is well correlated with a modified Eotvos number Eod which employs the maximum horizontal dimension of a deformed bubble as a characteristic length. An empirical correlation of CT was therefore summarized as a function of Re and Eod. The critical bubble diameter causing the radial void profile transition from wall peaking to core peaking in an air–water bubbly flow evaluated by the proposed CT correlation coincided with available experimental data.