The aerodynamics of hovering insect flight. V. A vortex theory

1984 ◽  
Vol 305 (1122) ◽  
pp. 115-144 ◽  
Keyword(s):  
Correction Factor ◽  
Insect Flight ◽  
Vortex Sheet ◽  
Hovering Flight ◽  
Vortex Theory ◽  
Actuator Disc ◽  
Temporal Pressure ◽  
The Mean ◽  
Wake Model ◽  
Induced Velocity

A full derivation is presented for the vortex theory of hovering flight outlined in preliminary reports. The theory relates the lift produced by flapping wings to the induced velocity and power of the wake. Suitable forms of the momentum theory are combined with the vortex approach to reduce the mathematical complexity as much as possible. Vorticity is continuously shed from the wings in sympathy with changes in wing circulation. The vortex sheet shed during a half-stroke convects downwards with the induced velocity field, and should be approximately planar at the end of a half-stroke. Vorticity within the sheet will roll up into complicated vortex rings, but the rate of this process is unknown. The exact state of the sheet is not crucial to the theory, however, since the impulse and energy of the vortex sheet do not change as it rolls up, and the theory is derived on the assumption that the extent of roll-up is negligible. The force impulse required to generate the sheet is derived from the vorticity of the sheet, and the mean wing lift is equal to that impulse divided by the period of generation. This method of calculating the mean lift is suitable for unsteady aerodynamic lift mechanisms as well as the quasi-steady mechanism. The relation between the mean lift and the impulse of the resulting vortex sheet is used to develop a conceptual artifice - a pulsed actuator disc - that approximates closely the net effect of the complicated lift forces produced in hovering. T he disc periodically applies a pressure impulse over some defined area, and is a generalized form of the Froude actuator disc from propeller theory. The pulsed disc provides a convenient link between circulatory lift and the powerful momentum and vortex analyses of the wake. The induced velocity and power of the wake are derived in stages, starting with the simple Rankine-Froude theory for the wake produced by a Froude disc applying a uniform, continuous pressure to the air. The wake model is then improved by considering a ‘modified’ Froude disc exerting a continuous, but non-uniform pressure. This step provides a spatial correction factor for the Rankine-Froude theory, by taking into account variations in pressure and circulation over the disc area. Finally, the wake produced by a pulsed Froude disc is analysed, and a temporal correction factor is derived for the periodic application of spatially uniform pressures. Both correction factors are generally small, and can be treated as independent perturbations of the Rankine-Froude model. Thus the corrections can be added linearly to obtain the total correction for the general case of a pulsed actuator disc with spatial and temporal pressure variations. The theory is compared with Rayner’s vortex theory for hovering flight. Under identical test conditions, numerical results from the two theories agree to within 3%. Rayner presented approximations from his results to be used when applying his theory to hovering animals. These approximations are not consistent with my theory or with classical propeller theory, and reasons for the discrepancy are suggested.

The aerodynamics of hovering insect flight. I. The quasi-steady analysis

1984 ◽  
Vol 305 (1122) ◽  
pp. 1-15 ◽  
Keyword(s):  
Insect Flight ◽  
Steady Motion ◽  
Hovering Flight ◽  
Kinematic Data ◽  
Vortex Theory ◽  
The Mean ◽  
Induced Velocity ◽  
Static Conditions ◽  
Unsteady Effects ◽  
Animal Flight

The conventional aerodynamic analysis of flapping animal flight invokes the ‘quasisteady assumption’ to reduce a problem in dynamics to a succession of static conditions: it is assumed that the instantaneous forces on a flapping wing are equivalent to those for steady motion at the same instantaneous velocity and angle of attack. The validity of this assumption and the importance of unsteady aerodynamic effects have long been controversial topics. Weis-Fogh tested the assumption for hovering animal flight, where unsteady effects are most pronounced, and concluded that most insects indeed hover according to the principles of quasi-steady aerodynamics. The logical basis for his conclusion is reviewed in this paper, and it is shown that the available evidence remains ambiguous. The aerodynamics of hovering insect flight are re-examined in this series of six papers, and a conclusion opposite to Weis-Fogh’s is tentatively reached. New morphological and kinematic data for a variety of insects are presented in papers II and III, respectively. Paper IV offers an aerodynamic interpretation of the wing kinematics and a discussion on the possible roles of different aerodynamic mechanisms. A generalized vortex theory of hovering flight is derived in paper V, and provides a method of estimating the mean lift, induced power and induced velocity for unsteady as well as quasi-steady flight mechanisms. The new data, aerodynamic mechanisms and vortex theory are all combined in paper VI for an analysis of the lift and power requirements and other mechanical aspects of hovering flight. A large number of symbols are needed for the morphological, kinematic and aerodynamic analyses. Most of them appear in more than one paper of the series, and so a single comprehensive table defining the major symbols from all of the papers is presented at the end of this paper.

Research of the Nonlinear Wake Model of HAWT

2013 ◽  
Vol 448-453 ◽  
pp. 1747-1753
Author(s):  
Rui Yang ◽  
Sheng Long Zhang ◽  
Jiu Xin Wang
Keyword(s):  
Radial Component ◽  
Wake Vortex ◽  
Surface Model ◽  
Trailing Edge ◽  
Vortex Sheets ◽  
Central Vortex ◽  
Actuator Disc ◽  
Horizontal Axis Wind Turbines ◽  
Wake Model ◽  
Induced Velocity

In the existing linear wake (cylindrical surface) model of horizontal axis wind turbines, the rotor was taken as the actuator-disc composed of infinite blades with infinitesimal chords. The distribution of variable circulation along blade was not taken into account and the span-wise (or radial) component of induced velocity is totally ignored. And assumed that the all trailing vortex filament shed from blade trailing edge would locate on their own cylindrical stream-surface. This aerodynamic model for determination of wake configuration is obviously different from that actually observed wake in wind tunnel experiment. Therefore, a "nonlinear" wake model was proposed, in this model the wake vortex system was divided into the central vortex along rotor axis, the bound vortex along blade axis, the wake vortex sheets shed from blade trailing edge and extent into infinity behind the rotor. Then, on the basis of potential theory in fluid mechanics a set of integral equations for evaluation of induced velocity in wake were derived with Biot-Savarts formula.

Lift and power requirements of hovering flight inDrosophila virilis

2002 ◽  
Vol 205 (16) ◽  
pp. 2413-2427 ◽  
Author(s):  
Mao Sun ◽  
Jian Tang
Keyword(s):  
Flight Control ◽  
Stokes Equations ◽  
Fruit Fly ◽  
The Body ◽  
Drosophila Virilis ◽  
Specific Power ◽  
Aerodynamic Forces ◽  
Hovering Flight ◽  
Power Expenditure ◽  
The Mean

SUMMARYThe lift and power requirements for hovering flight in Drosophila virilis were studied using the method of computational fluid dynamics. The Navier-Stokes equations were solved numerically. The solution provided the flow velocity and pressure fields, from which the unsteady aerodynamic forces and moments were obtained. The inertial torques due to the acceleration of the wing mass were computed analytically. On the basis of the aerodynamic forces and moments and the inertial torques, the lift and power requirements for hovering flight were obtained.For the fruit fly Drosophila virilis in hovering flight (with symmetrical rotation), a midstroke angle of attack of approximately 37°was needed for the mean lift to balance the insect weight, which agreed with observations. The mean drag on the wings over an up- or downstroke was approximately 1.27 times the mean lift or insect weight (i.e. the wings of this tiny insect must overcome a drag that is approximately 27 % larger than its weight to produce a lift equal to its weight). The body-mass-specific power was 28.7 W kg-1, the muscle-mass-specific power was 95.7 W kg-1 and the muscle efficiency was 17 %.With advanced rotation, larger lift was produced than with symmetrical rotation, but it was more energy-demanding, i.e. the power required per unit lift was much larger. With delayed rotation, much less lift was produced than with symmetrical rotation at almost the same power expenditure; again, the power required per unit lift was much larger. On the basis of the calculated results for power expenditure, symmetrical rotation should be used for balanced, long-duration flight and advanced rotation and delayed rotation should be used for flight control and manoeuvring. This agrees with observations.

A modified near wake dynamic model for rotor analysis

1999 ◽  
Vol 103 (1021) ◽  
pp. 143-146 ◽  
Author(s):  
T. Wang ◽  
F. N. Coton
Keyword(s):  
High Resolution ◽  
Dynamic Model ◽  
Numerical Evaluation ◽  
Helicopter Rotor ◽  
Vortex Interaction ◽  
Near Wake ◽  
Blade Vortex Interaction ◽  
Wake Model ◽  
Induced Velocity

Abstract The Beddoes near wake model, developed for high resolution blade vortex interaction computations, enables efficient numerical evaluation of the downwash due to trailed vorticity in the near wake of a helicopter rotor. The model is, however, limited by the assumption that the near wake lies in the plane of the rotor and, in some cases, by its inability to accurately evaluate the induced velocity contribution from vorticity trailed from inboard blade sections. In this paper, modifications to the method are proposed which address these issues and allow it to be used with confidence over a wider range of rotor flows.

A Note on the Design of Wake-Adapted Propellers

1980 ◽  
Vol 24 (04) ◽  
pp. 227-231
Author(s):  
Gilbert Dyne
Keyword(s):  
Vortex Theory ◽  
The Difference ◽  
Wake Model ◽  
Effective Wake ◽  
Leveling Effect

Problems associated with the design of wake-adapted propellers are illustrated by results obtained from a simple propeller and wake model. It is shown that the widely used approach of the vortex theory gives much-too-low induced axial velocities at the inner radii of the propeller, which results in too-low blade pitch ratios. The reason for this is that no regard is paid to the fact that the vorticity of the approaching flow is changed by the propeller. If this changing vorticity is introduced in the vortex theory, its shortcomings are eliminated. An effect of the improvement is that the effective wake at the propeller plane differs from the nominal wake. The difference depends upon the propeller load. The propeller is shown to have a leveling effect on a nonuniform axisymmetric nominal wake.

Renal excretion of thiamin by the dog

1960 ◽  
Vol 198 (6) ◽  
pp. 1274-1278 ◽  
Author(s):  
Gerhard Malnic ◽  
Alberto Carvalho da Silva ◽  
Rebecca C. de Angelis ◽  
Zulmira J. Gomes
Keyword(s):  
Protein Binding ◽  
Creatinine Clearance ◽  
Correction Factor ◽  
Plasma Levels ◽  
Renal Excretion ◽  
Extraction Ratio ◽  
Renal Tubules ◽  
Back Diffusion ◽  
Water Reabsorption ◽  
The Mean

Simultaneous thiamin and creatinine clearance determinations in unanesthetized dogs revealed a pattern of excretion characteristic of a substance actively excreted by the renal tubules. The maximum tubular excretory capacity per minute (Tm) calculated at high thiamin plasma levels was negative, unless a correction factor (FW) of 0.63, calculated from the clearance values, was applied. Under these conditions, the mean Tm value was 1463 µg/min. By means of perfusion experiments on isolated dog kidneys it was shown that there was very little or no protein binding of thiamin. In extraction-ratio determinations on unanesthetized dogs it was shown that thiamin was not destroyed by the kidney and that the extraction of thiamin and PAH were similar. At varying urine flows with high thiamin plasma levels, a correlation between water reabsorption and thiamin excretion could be observed, suggesting the occurrence of back-diffusion of thiamin under these conditions.

A numerical investigation of the rolling-up of vortex sheets

1980 ◽  
Vol 373 (1752) ◽  
pp. 67-91 ◽  
Keyword(s):  
Numerical Integration ◽  
Numerical Investigation ◽  
Equations Of Motion ◽  
Additional Term ◽  
Vortex Sheet ◽  
Time Step ◽  
Vortex Sheets ◽  
Numerical Instabilities ◽  
Sinusoidal Disturbance ◽  
Induced Velocity

In this paper the development of a vortex sheet due to an initially sinusoidal disturbance is calculated. When determining the induced velocity in points of the vortex sheet, it can be represented by concentrated vortices but it is shown that it is analytically more correct to add an additional term that represents the effect of the immediate neighbourhood of the point considered. The equations of motion were integrated by a Runge-Kutta technique to exclude numerical instabilities. The time step was determined by the requirement that a quantity (Hamiltonian) that remains invariant as a result of the equations of motion, should not change more than a certain amount in the numerical integration of the equations of motion. One difficulty is that if a greater number of concentrated vortices are introduced to represent the vortex sheet, the effect of round-off errors becomes more important. The number of figures retained in the computations limits the number of concentrated vortices. Where the round-off errors have been kept sufficiently small, a process of rolling-up of vorticity clearly occurs. There is no point in pursuing the calculations much beyond this point, first because the representation of the vortex sheet by concentrated vortices becomes more and more inaccurate and secondly because viscosity will have the effect of transforming the rolled-up vortex sheet into a region of vorticity.

scholarly journals Models of internal jumps and the fronts of gravity currents: unifying two-layer theories and deriving new results

2018 ◽  
Vol 846 ◽  
pp. 654-685 ◽  
Author(s):  
Marius Ungarish ◽  
Andrew J. Hogg
Keyword(s):  
Velocity Field ◽  
Control Volume ◽  
Gravity Current ◽  
Vortex Sheet ◽  
Gravitational Acceleration ◽  
Two Fluids ◽  
Dissipative Effects ◽  
Wake Model ◽  
New Formulation ◽  
Vorticity Balance

The steady speeds of the front of a gravity current and of an internal jump on a two-layer stratification are often sought in terms of the heights of the relatively dense fluid both up- and downstream from the front or jump, the height of the channel within which they flow, the densities of the two fluids and gravitational acceleration. In this study a unifying framework is presented for calculating the speeds by balancing mass and momentum fluxes across a control volume spanning the front or jump and by ensuring the assumed pressure field is single-valued, which is shown to be equivalent to forming a vorticity balance over the control volume. Previous models have assumed the velocity field is piecewise constant in each layer with a vortex sheet at their interface and invoked explicit or implicit closure assumptions about the dissipative effects to derive the speed. The new formulation yields all of the previously presented expressions and demonstrates that analysing the vorticity balance within the control volume is a useful means of constraining possible closure assumptions, which is arguably more effective than consideration of the flow energetics. However the new approach also reveals that a novel class of models may be developed in which there is shear in the velocity field in the wake downstream of the front or the jump, thus spreading the vorticity over a layer of non-vanishing thickness, rather than concentrating it into a vortex sheet. Mass, momentum and vorticity balances applied over the control volume allow the thickness of the wake and the speed of the front/jump to be evaluated. Results from this vortex-wake model are consistent with published numerical simulations and with data from laboratory experiments, and improve upon predictions from previous formulae. The results may be applied readily to Boussinesq and non-Boussinesq systems and because they arise as simple algebraic expressions, can be straightforwardly incorporated as jump conditions into spatially and temporally varying descriptions of the motion.

scholarly journals Parameterization of Mixed Layer and Deep-Ocean Mesoscales including Nonlinearity

2018 ◽  
Vol 48 (3) ◽  
pp. 555-572 ◽  
Author(s):  
V. M. Canuto ◽  
Y. Cheng ◽  
M. S. Dubovikov ◽  
A. M. Howard ◽  
A. Leboissetier
Keyword(s):  
Deep Ocean ◽  
Altimetry Data ◽  
Dynamical Equations ◽  
Mean Velocity ◽  
Ocean Stratification ◽  
Large Heat ◽  
Reference Experiment ◽  
The Mean ◽  
The Difference ◽  
Induced Velocity

AbstractIn 2011, Chelton et al. carried out a comprehensive census of mesoscales using altimetry data and reached the following conclusions: “essentially all of the observed mesoscale features are nonlinear” and “mesoscales do not move with the mean velocity but with their own drift velocity,” which is “the most germane of all the nonlinear metrics.” Accounting for these results in a mesoscale parameterization presents conceptual and practical challenges since linear analysis is no longer usable and one needs a model of nonlinearity. A mesoscale parameterization is presented that has the following features: 1) it is based on the solutions of the nonlinear mesoscale dynamical equations, 2) it describes arbitrary tracers, 3) it includes adiabatic (A) and diabatic (D) regimes, 4) the eddy-induced velocity is the sum of a Gent and McWilliams (GM) term plus a new term representing the difference between drift and mean velocities, 5) the new term lowers the transfer of mean potential energy to mesoscales, 6) the isopycnal slopes are not as flat as in the GM case, 7) deep-ocean stratification is enhanced compared to previous parameterizations where being more weakly stratified allowed a large heat uptake that is not observed, 8) the strength of the Deacon cell is reduced. The numerical results are from a stand-alone ocean code with Coordinated Ocean-Ice Reference Experiment I (CORE-I) normal-year forcing.

A computational model of the flight dynamics and aerodynamics of a jellyfish-like flying machine

2017 ◽  
Vol 819 ◽  
pp. 621-655 ◽  
Author(s):  
Fang Fang ◽  
Kenneth L. Ho ◽  
Leif Ristroph ◽  
Michael J. Shelley
Keyword(s):  
Aerodynamic Force ◽  
Fast Multipole Method ◽  
Flow Simulation ◽  
Ground Effect ◽  
Vortex Sheet ◽  
Mirror Image ◽  
Design Parameters ◽  
Efficiency Increase ◽  
Wake Model ◽  
Opening And Closing

We explore theoretically the aerodynamics of a recently fabricated jellyfish-like flying machine (Ristroph & Childress, J. R. Soc. Interface, vol. 11 (92), 2014, 20130992). This experimental device achieves flight and hovering by opening and closing opposing sets of wings. It displays orientational or postural flight stability without additional control surfaces or feedback control. Our model ‘machine’ consists of two mirror-symmetric massless flapping wings connected to a volumeless body with mass and moment of inertia. A vortex sheet shedding and wake model is used for the flow simulation. Use of the fast multipole method allows us to simulate for long times and resolve complex wakes. We use our model to explore the design parameters that maintain body hovering and ascent, and investigate the performance of steady ascent states. We find that ascent speed and efficiency increase as the wings are brought closer, due to a mirror-image ‘ground-effect’ between the wings. Steady ascent is approached exponentially in time, which suggests a linear relationship between the aerodynamic force and ascent speed. We investigate the orientational stability of hovering and ascent states by examining the flyer’s free response to perturbation from a transitory external torque. Our results show that bottom-heavy flyers (centre of mass below the geometric centre) are capable of recovering from large tilts, whereas the orientation of the top-heavy flyers diverges. These results are consistent with the experimental observations in Ristroph & Childress (J. R. Soc. Interface, vol. 11 (92), 2014, 20130992), and shed light upon future designs of flapping-wing micro aerial vehicles that use jet-based mechanisms.

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