Kutta Joukowski Research Articles

Единая теория движителей на непрерывных потоках. Краткая теория однорядных воздушных винтов

The main mistake that Professor Nikolay Yegorovich Joukowski has committed in formulating the Kutta–Joukowski theorem , formula (1), is the introduction of such an incorrect notion as circulation, . The Kutta–Joukowski theorem , derived by Professor Nikolay Yegorovich Joukowski , does not stand up to any criticism, as it absolutely does not agree with experimental data of the modern theory of jet engines, where it, formula (1), is incorporated as fundamental. This is due to the fact that under such a concept as circulation, , the maximum pressure of blowing flow on the blade of blowing profile is where circulation is maximum, that is on the outlet edge of blowing profile, and zero pressure of blowing flow on the blade of blowing profile is on the inlet edge, as on the inlet edge .Such a diagram of pressures of blowing flow on the blade of blowing profile, generating a gradient of static pressures, directed against the flow, is in direct contradiction with the experimental data of modern theory of jet engines, which declare that the maximum pressure of blowing flow on the blade of blowing profile is within the inlet edge and this pressure is gradually reduced as blowing flow is displaced to the outlet edge of blowing profile, thus forming the pressure gradient directed along the flow, [1]. Therefore, the Kutta–Joukowski theorem , derived by Professor Nikolay Yegorovich Joukowski in 1912, is incorrect and can not be fundamental for any theory, this theorem should be formulated in the other way, which is described in the paper.

Low Reynolds number airfoil aerodynamic loads determination via line integral of velocity obtained with particle image velocimetry

The small magnitude lift forces generated by both a NACA 0012 airfoil and a thin flat plate at Re = 29,000 and 54,000 were determined through the line integral of velocity, obtained with particle image velocimetry, via the application of the Kutta–Joukowsky theorem. Surface pressure measurements of the NACA0012 airfoil were also obtained to validate the lift coefficient C l. The bound circulation was found to be insensitive to the size and aspect ratio of the rectangular integration loop for pre-stall angles. The present C l data were also found to agree very well with the surface pressure-determined lift coefficient for pre-stall conditions. A large variation in C l with the loop size and aspect ratio for post-stall conditions was, however, observed. Nevertheless, the present flat-plate C l data were also found to collectively agree with the published force-balance measurements at small angles of attack, despite the large disparity exhibited among the various published data at high angles. Finally, the ensemble-averaged wake velocity profiles were also used to compute the drag coefficient and, subsequently, the lift-to-drag ratio.

Kutta–Joukowsky theorem in viscous and unsteady flow

Nominally two-dimensional air flow over a thin flat plate at low Reynolds number is investigated. The primary objective is to experimentally determine with good accuracy the small magnitude lift force, generated by the plate at various angles of attack, by means of application of the Kutta–Joukowsky theorem where circulation is obtained from the line integral of velocity around the flat plate using non-invasive laser doppler velocimeter. Specific focus is on assessing applicability of the Kutta–Joukowsky theorem, originally theorized for inviscid and steady flow, in the post-stall region. At high angles of attack, due to severe flow separation from both the edges of the flat plate and occurrence of periodic vortex shedding, wake flow is found to be highly viscous, turbulent and unsteady. Nevertheless, the results show a remarkably good agreement with previous investigations in both the linear range and the non-linear range of the lift curve without any correction applied to the data. The line integral of velocity along the rectangular loop enclosing the flat plate shows that the vertical components, albeit smaller in magnitude, possess the same sign and hence are additive in contribution to the circulation, whereas the horizontal components possess opposite signs and hence are subtractive in their contribution to the circulation. The paper presents some interesting and hitherto undisclosed features of flow field around the flat plate.

Diffraction of an impulsive line source with wake

The problem of diffraction due to an impulse line source by an absorbing half-plane with wake using Myres' impedance condition (Myers 1980 J. Sound Vib.71 429–34) in the presence of a subsonic fluid flow is studied. The time dependence of the field requires a temporal Fourier transform in addition to the spatial Fourier transform. The solution of the problem in the presence of wake is obtained by using Greens' function method, Fourier transform, the Wiener–Hopf technique and the modified stationary phase method. Expressions for the total far field for the trailing edge (wake present) situation are given. It is observed that the field produced by the Kutta–Joukowski condition will be substantially in excess of the field when this condition is ignored. Finally, a simple procedure is devised to calculate the inverse temporal Fourier transform. The solution for the leading edge situation can be obtained if the wake, and consequently a Kutta–Joukowski edge condition, is ignored. This can also be seen from the numerical results.

Application of the actuator surface concept to wind turbine rotor aerodynamics

AbstractThe structure of blade tip vortices is recognized as a key issue in wind turbine aerodynamic modelling by many researchers in the field. In the search for an intermediate model between full Navier–Stokes and blade‐element momentum simulations, this article presents a method using rotating actuator surfaces to model wind turbine aerodynamics. An actuator surface is a simple planar surface, porous to the flow, which is characterized by velocity and pressure discontinuities, whose action on the flow is achieved through an attached system of forces. These discontinuities and forces are determined from blade‐element analysis and the Kutta–Joukowski relation. After implementing this concept in a three‐dimensional CFD (Computational Fluid Dynamics) method, results are produced for the experimental rotors of NREL and TUDelft. The method is validated against both experimental measurements and the predictions of three other numerical models for wind turbine aerodynamic analysis. Qualitative and quantitative comparisons show that the actuator surface concept agrees well with the other numerical models. In addition to rotor aerodynamic analysis, the actuator surface concept can be used in the study of wake aerodynamics, or as the Eulerian flow solver in hybrid methods. Copyright © 2009 John Wiley & Sons, Ltd.

The unsteady pressure field and the aerodynamic performances of a Savonius rotor based on the discrete vortex method

The aim of this paper is to numerically explore the non-linear two-dimensional unsteady potential flow over a Savonius rotor and to develop a code for predicting its aerodynamics performances. In the model developed, the rotor is represented in a median plane by two semicircles, displaced along their common diameter. The two semicircles can be considered to produce lifting effects. As a result, they are modelled by a collection of discrete vortices on their contours. The flow field is then governed by the Laplace equation. The versatile Neumann boundary condition, applied over the contour of the semicircles and the Kutta Joukowsky condition applied at the four extremities of the semicircles have been used in the modelling. The torque distribution of the stationary rotor and the unsteady pressure field on the blades of the rotating rotor, predicted by the code developed, have been compared and validated by some experimental data.

Particle image velocimetry investigation of an aerofoil with a Gurney flap

The current paper reviews the research on NACA 23012 and FX66-17AII-182 aerofoils, aerodynamic performance for different angles of attack at Reynolds number Re ≍ 3 × 105, based on chord, both with clean configuration and the flaps. Flow field around the aerofoil was measured using a two-dimensional particle image velocimetry (PIV) system, and the measurement of static pressure on aerofoil surface was carried out. The current article also describes new methods in Gurney flap research, using the two-dimensional PIV system and Kutta—Joukowski lift theorem or static pressure assumption to evaluate aerofoil lift.

On the Circulation and the Position of the Forward Stagnation Point on Airfoils

Lift and velocity circulation around airfoils are two aspects of the same phenomenon when airfoils are not stalled and the Kutta—Joukowski theorem applies. This theorem establishes a linear dependence between lift and circulation, which breaks when stalling occurs. As the angle of attack increases beyond this point, the circulation vanishes. Since the circulation determines to a great extent the position of the forward stagnation point on an airfoil, the measurement of this position is an easy and simple way to determine the circulation, which is of help in understanding the role of the latter in the generation of aerodynamic forces on airfoils.

Modeling Response of Flexible High-Aspect-Ratio Wings to Wind Turbulence

Analytical modeling of the effect of wind turbulence on flexible high-aspect-ratio aircraft wings typical of unmanned air vehicles (UAVs). The wind model is derived from the Kolmogorov power-law spectrum random field turbulence model invoking the Taylor “frozen-field” hypothesis. The aerodynamic model is based on the typical-section compressible attached-flow with Kutta–Joukowski boundary conditions. The gust loading—both lift and moment—is calculated explicitly for M=0 and M=1 as typical of subsonic and transonic flow. The gust loading intensity is shown to decrease as the speed increases, so that the turbulence effects are not significant at transonic speeds. To calculate the wing response we use the continuum two degree-of-freedom cantilever beam model of Goland and derive explicit expressions of the spectral density of both the plunge (bending) and pitch (torsion) response for M=0 . Numerical results are presented for two illustrative wings. Most of the turbulence energy is in the 0–10H range. Since f...

DIFFRACTION OF AN ACOUSTIC WAVE BY A PLATE IN A UNIFORM FLOW: A NUMERICAL APPROACH

We study the diffraction in time harmonic regime of an acoustic wave by a rigid plate in the presence of a uniform flow in a duct. Contrary to prior analytical studies, using Wiener–Hopf techniques and thus restricted to semi-infinite plates, we use a finite elements method which allows us to deal with plates of finite length. To take into account irrotational perturbations induced by the trailing edge of the plate, a potential formulation requires the introduction of a vortex sheet behind the plate. The key point of the method is to get access at the singular coefficient of the velocity potential near the trailing edge, in order to cancel it using the so-called Kutta–Joukowski condition. This approach leads to an efficient finite elements method, and numerical computations are presented: we show the amplitude of the vortex sheet versus the Mach number and the plate length and the dissipated acoustic power versus the Mach number and the frequency. This method is extended to the case of two aligned plates to analyze the influence of the choice of the boundary condition on the downstream plate which interacts with a vortex sheet.

Diffraction by a half-plane in a moving fluid

In the following work we solve the problem of the diffraction of a plane sound wave by an impedance half-plane in a moving fluid. Expressions for the total far field are derived for both the leading edge and trailing edge situations. In the trailing edge situation the problem has the added complication of a trailing vortex sheet or wake. Hence a Kutta–Joukowski edge condition is imposed to ensure that the fluid velocity is finite at the edge and to obtain a unique solution to the problem.

Ultrasonic Lift Measurement Technique for Flow-Induced Structural Vibrations

Deficiencies exist in our current ability to measure lift forces in wind-tunnel experiments on vibrating structures in a fluid flow. The ultrasonic lift measurement (ULM) technique has been previously developed to measure time-averaged fluid circulation and lift on stationary structures. The ULM technique is based on measuring transit times of acoustic pulses along paths enclosing the structure. A quasi-steady method based on the Kutta–Joukowski theorem has been used in the past to convert fluid circulation to lift values in ULM studies. In this paper, the largely unstudied extension of the ULM technique to measure unsteady lift forces in flows involving structural vibration is considered. Analytic methods are developed that can be used to properly convert the instantaneous circulation measurements (attainable from ULM experiments) to lift values. These unsteady methods are validated by using numerical simulations of flows over flat plates undergoing oscillating motion. It is shown that the addition of un...

Two-dimensional potential theory for translating and rotating solids

Two-dimensional inviscid theory is adapted to evaluate easily the flow, hydrodynamic force and moment of the force on a solid that translates and rotates with rather arbitrary imposed velocities. As an example a moving thin Kutta–Joukowski airfoil is considered.

Scattering of a Spherical Gaussian Pulse Near an Absorbing Half Plane

The space–time acoustic wave diffraction due to a spherical Gaussian pulse near an absorbing half plane introducing the Kutta–Joukowski condition (wake condition) is considered. The temporal Fourier transform is used to calculate the diffracted field. It is found that the field produced by the Kutta–Joukowski condition will be substantially in excess of that in its absence when the source is near the edge.

Sub-Mach-1 sound from a point source near an absorbing half-plane

The scattering of an acoustic wave from a point source near an absorbing half-plane introducing the Kutta–Joukowski condition is studied. The effect of absorption on the diffracted field is shown explicitly. It is found that the field produced by the Kutta–Joukowski condition will be substantially larger than the field produced in its absence when the source is near the edge. It is further observed that the Kutta–Joukowski condition and the absorption do not have much influence on the scattered field away from the diffracting plane.

Stability and convergence of streamline curvature flow analysis procedures

AbstractA convergence technique for use with the streamline curvature method of cascade and channel flow is presented which produces stable and rapid convergence for subsonic, transonic and supersonic flow and also in the downstream region where the Kutta–Joukowski condition is applied to a cascade flow. The convergence method is fully two‐dimensional in nature, the iterative changes being determined via a model which takes into account the actual streamwise error distribution and the real normal changes in the grid.

The motion of a vortex filament with axial flow

Infinitesimal waves on a uniform vortex with axial flow are studied. The equation for the frequency of helical waves is obtained, and solved for the case of long waves which leave the internal structure almost unaltered. A method is developed to obtain results for vortices of non-uniform structure and for displacements which are not necessarily small compared with the core radius. The approach consists of balancing the Kutta—Joukowski lift force, the momentum flux due to the axial motion, and the ‘tension’ of the vortex lines. A general equation for the motion of a vortex filament is obtained, valid for arbitrary shape and internal structure, and in the presence of an external irrotational velocity field. When the axial flow vanishes, the method is equivalent to using the Biot—Savart law for the self-induced velocity, with a suitable cutoff. The impulse of a vortex filament is discussed and its rate of change is given.

Slender-body analysis of the motion and stability of a vortex filament containing an axial flow

Previous results concerning the effects of axial velocity on the motion of vortex filaments are reviewed. These results suggest that a slender-body force balance between the Kutta–Joukowski lift on the vortex cross-section and the momentum flux within the curved filament will give some insight into the behaviour of the filament. These simple ideas are exploited for both a single vortex filament and a vortex pair, both containing axial flow. The stability of a straight vortex filament containing an axial flow to long wave sinusoidal displacements of its centre-line is investigated and the stability boundary obtained. The effect of axial flow on the stability of a vortex pair is explored. It is shown that to lowest order (in the ratio of vortex core radius to distance between the vortices) the effect of axial flow is to reduce the self-induced rotation of a single filament and that this effect can be considered as a change in effective core radius. To the next order, travelling waves appear in the instability, the instability mode for the vortex pair becomes non-planar but the amplification rate of the instability is not affected.

Streamline Curvature Analysis of Compressible and High Mach Number Cascade Flows

A blade to blade design tool for analysing the subsonic, transonic and supersonic flows in axial turbine blades is presented using the streamline curvature method. The work includes a numerical convergence analysis which produced convergence of the digital computer programme for all cases attempted. Computed results are compared with experiment and other theory. Comparisons for turbine type cascades up to Mach 10 show a satisfactory agreement and techniques introduced give improved leading edge results and allow the Kutta–Joukowski condition to be applied at the trailing edge for the accurate determination of the velocity distribution and the cascade deviation. Predictions for convergent-divergent supersonic wind tunnel type nozzles agree closely with experiment and with other theories. Numerical difficulties are found to be more acute for subsonic/supersonic turbine cascades and experimental agreement, while acceptable, is inferior to that obtained for subsonic cascades.

Wake curvature and the Kutta condition

The potential problem for the flow at high Reynolds numbers R outside the boundary layer and wake of a thin flat plate at small incidence with allowance for displacement thickness is not fully denned unless the position of the wake is known in advance. The Kutta–Joukowski hypothesis does not provide a satisfactory first approximation to this because of the singularity in curvature of the streamline springing from the trailing edge in inviscid flow, which implies that the initial curvature of the wake in the real flow will be large enough to cause a modification to the potential flow. The net vorticity per unit length in a curved wake is approximately U∞ δ2∞dθ0/ds, where U∞, δ2∞ and dθ0/ds are respectively the undisturbed stream velocity, momentum thickness at infinity and curvature. The outer potential problem is set up with a vortex sheet of this strength to represent the wake, leading to a singular integro-differential equation for θ0(s). From the particular solution we obtain a proportionate correction – (CD/4π) (log4/CD) to the Kutta–Joukowski circulation, where CD is the drag coefficient. For laminar flow this is of order R−½ log (1/R). However, the solution also contains an arbitrary constant which cannot be settled without an examination of the near wake. The recent work of Brown & Stewartson (1970) suggests that this may lead to a term of the lower order , but depends on the assumption, not supported by the present analysis, that the pressure rise across the wake is o(R−¼).