Joukowski Aerofoil Research Articles

The starting vortices generated by bodies with sharp and straight edges in a viscous fluid

A two-dimensional body that moves suddenly in a viscous fluid can instantly generate vortices at its sharp edges. Recent work using inviscid flow theory, based on the Birkhoff–Rott equation and the Kutta condition, predicts that the ‘starting vortices’ generated by the sharp and straight edges of a body – i.e. the vortices formed immediately after motion begins – can be one of three distinct self-similar types. We explore the existence of these starting vortices for a flat plate and two symmetric Joukowski aerofoils immersed in a viscous fluid, using high-fidelity direct numerical simulations (DNS) of the Navier–Stokes equations. A lattice Green's function method is employed and simulations are performed for chord Reynolds numbers ranging from 5040 to 45 255. Vortices generated at the leading and trailing edges of the flat plate show agreement with the derived inviscid theory, for which a detailed assessment is reported. Agreement is also observed for the two symmetric Joukowski aerofoils, demonstrating the utility of the inviscid theory for arbitrary bodies. While this inviscid theory predicts an abrupt transition between the starting-vortex types, DNS shows a smooth transition. This behaviour occurs for all Reynolds numbers and is related to finite-time effects – there is a maximal time for which the (self-similar) starting vortices exist. We confirm the inviscid prediction that the leading-edge starting vortex of a flat plate can be suppressed dynamically. This has implications for the performance of low-speed aircraft such as model aeroplanes, micro air vehicles and unmanned air vehicles.

Broadband noise prediction for aerofoils with a serrated trailing edge based on Amiet’s theory

Trailing-edge serrations are used as means to reduce aerofoil broadband self-noise in various industrial applications. However, a reliable prediction of the reduced noise level has not been established to date. In this paper, the authors suggest that Amiet’s theory can be used to accurately predict the sound radiated from serrated trailing edges. A plausible explanation as to how the classical theory, originally derived from non-serrated (straight) trailing edges, also works for the serrated trailing edges is provided. Besides, the authors propose a directivity correction to the Amiet’s original model to account for realistic aerofoil geometry. The directivity correction provides a significantly improved prediction of the noise level in the upstream and downstream directions for both serrated and baseline trailing edge cases. This study is based on wall-resolved large-eddy simulations of Joukowski aerofoils with multiple serration geometries and in different flow conditions.

Effect of serrated trailing edges on aerofoil tonal noise

Abstract

On high-frequency sound generated by gust–aerofoil interaction in shear flow

AbstractA theoretical model is constructed to predict the far-field sound generated by high-frequency gust–aerofoil interaction in steady parallel shear flow, including the effects of aerofoil thickness. Our approach is to use asymptotic analysis of the Euler equations linearised about steady parallel shear flow, in the limits of high frequency and small, but non-zero, aerofoil thickness and Mach number. The analysis splits the flow into various regions around the aerofoil; local inner regions around the leading and the trailing edges where sound is generated and scattered; a surface transition region accounting for the curvature of the aerofoil; a wake transition region downstream of the aerofoil; and an outer region through which the sound propagates to the observer. Solutions are constructed in all regions, and matched using the principle of matched asymptotic expansions to yield the first two terms in the expansion of both the amplitude and the phase of the far-field pressure. Result are computed for the particular case of scattering of a gust by a symmetric Joukowski aerofoil placed in symmetric Gaussian parallel shear flow. The introduction of mean shear is shown to have a significant effect on the far-field directivity and on the total radiated power.

Indirect Boundary Element Method for Calculation of Inviscid Compressible Flow Past a Joukowski Aerofoil with Linear Element Approach Using Doublet Distribution Alone

In this paper, an indirect boundary element method is used to give solution for surface as well as ground water bodies which are major nutrition fluids. The inviscid compressible flow (i.e., velocity distribution) over the surface of the joukowski aerofoil has been calculated with linear element approach using doublet distribution alone whereas in our previous research papers, we applied constant boundary element approach for this purpose. To check the accuracy of the method, the computed flow velocity is compared with the exact velocity. The comparison of these results has been given in the tables and graphs. It is found that the computed results are in good agreement with the analytical results.

Complex mapping of aerofoils – a different perspective

In this article an application of conformal mapping to aerofoil theory is studied from a geometric and calculus point of view. The problem is suitable for undergraduate teaching in terms of a project or extended piece of work, and brings together the concepts of geometric mapping, parametric equations, complex numbers and calculus. The Joukowski and Karman–Trefftz aerofoils are studied, and it is shown that the Karman–Trefftz aerofoil is an improvement over the Joukowski aerofoil from a practical point of view. For the most part only a spreadsheet program and pen and paper is required, only for the last portion of the study of the Karman–Trefftz aerofoils a symbolic computer package is employed. Ignoring the concept of a conformal mapping and instead viewing the problem from a parametric point of view, some interesting mappings are obtained. By considering the derivative of the mapped mapping via the chain rule, some new and interesting analytical results are obtained for the Joukowski aerofoil, and numerical results for the Karman–Trefftz aerofoil.

The Absolute Instability of Thin Wakes in an Incompressible/Compressible Fluid

RID=""ID="" Communicated by P. HallAbstract:The absolute/convective instability of two-dimensional wakes forming behind a flat plate and near the trailing-edge of a thin wedge-shaped aerofoil in an incompressible/compressible fluid is investigated. The mean velocity profiles are obtained by solving numerically the classical compressible boundary-layer equations with a negative pressure gradient for the flat plate case, and the incompressible triple-deck equations for a thin wedge-shaped trailing-edge. In addition for a Joukowski aerofoil the incompressible mean boundary-layer flow in the vicinity of the trailing-edge is also calculated by solving the interactive boundary-layer equations. A linear stability analysis of the boundary-layer profiles shows that a pocket of absolute instability occurs downstream of the trailing-edge with the extent of the instability region increasing with more adverse pressure gradients. The region of absolute instability persists along the near-wake axis, while the majority of the wake is convectively unstable. For a thin wedge-shaped trailing-edge in an incompressible fluid, a similar stability analysis of the velocity profiles obtained via a composite expansion, also shows the occurrence of absolute instability behind the trailing-edge for a wedge angle greater than a critical value. For increasing values of the wedge angle and for thicker aerofoils, separation takes place near the trailing-edge and the extent of absolute instability increases. Calculations also show that for insulated plates compressibility has a stabilizing effect but cooling the wall destabilizes the flow unlike wall heating.}

Numerical simulation of the effect of a moving wall on separation of flow past a symmetrical aerofoil

This paper applies the numerical simulation technique based on the generalized conservation of circulation (GCC) method to investigate the effects of a leading-edge rotating circular cylinder on the suppression of stall flow past a symmetrical Joukowski aerofoil. The variables investigated were the angle of attack α and the ratio of the surface velocity of the cylinder to freestream velocity, CU. The Reynolds number based on chord length is 1.43 × 105. It was found that the separation point on the upper surface of the aerofoil shifts downstream with increasing CU and stall flow can be significantly suppressed even at α up to 30° when CU = 4. The lift coefficient CL increases and the drag coefficient Cd decreases with increasing CU and the optimum CL/ Cd occurs at α=80°. The maximum CL/ Cd obtained is about 60 at CU = 4.

Aerofoil ground effect revisited

AbstractSteady state aerofoil ground effect is studied both numerically and analytically. Discrete vortex and linear vortex panel methods are applied to a parabolic arc and symmetric Joukowski aerofoil, respectively. A single vortex model for the flow over the parabolic arc aerofoil is developed. The single vortex model and other analytical solutions, valid either near or far from the ground, are compared with the numerical results. The numerical results are used to delineate the influences of angle of attack, camber and thickness. For small values of camber and angle of attack, normalised lift is enhanced near the ground and reduced far from it. For a fixed distance above the ground, normalised lift decreases with increasing angle of attack and camber. Thickness reduces lift at all heights above the ground.

Prandtl-Batchelor flow on a circular cylinder and on aerofoil sections

AbstractSteady, inviscid, incompressible, two dimensional flows containing vortex patches bounded by vortex sheets are used to model finite separated regions on certain closed bodies as an approach to modelling separation bubbles. First of all the problem of a vortex patch on a circle is considered. Calculations were done for symmetric flows covering a wide range of values of the separation position, the strength of the vortex sheet and the circulation at infinity and for non-symmetric flows at various angles of incidence. Smooth separation was detected in some cases. This model is used as an intermediate step for the calculation of the flow around an ellipse and a Joukowski aerofoil with the use of the Joukowski transformation.

Boundary-layer control by heat and mass transfer

Slot injection into the flow over a Joukowski aerofoil is considered. A narrow slot is placed close to the leading edge and the effect of injection of fluid from this on the boundary layer over the aerofoil is examined. In particular it is shown how separation may be delayed and, when cold fluid is injected, how the surface downstream from separation may be cooled.

Accurate finite element method for steady and quasisteady lifting aerofoil

A finite element formulation based on superposition is proposed for a lifting aerofoil in incompressible potential flow. An accuracy improvement technique for the singularity at the aerofoil trailing edge is discussed. It is shown that the quarter-node quadrilateral element can be readily employed to simulate this singularity. It is also demonstrated that the circulation in the flow field can be easily represented by a single constraint equation rather than introducing an artificial cut line in the mesh. The influence of finite-element mesh size on solution accuracy and the correct form of the boundary condition have been investigated as well. Numerical examples are given for both steady and quasisteady Joukowski aerofoils of various thicknesses and at a range of incidences. In all test cases, good agreement is observed between the analytical solution and the numerical result.

B108 動画表示を用いた流体力学教育用プログラム例

It is useful in education on fluid dynamics to show animated computer graphics expressed some kinds of flows, since they are dynamic phenomena. Therefore some animation programs by means of tracer method in which tracer points along streamline were successively illuminated by changing color codes for palettes were developed in this report. They were applied to potential flow around a rotating cylinder and around a joukowski aerofoil, Pitot tube in a flow, flow in an inlet section of a pipe, vortex flow, laminar and turbulent flow in a pipe.

Prediction of oscillating thick cambered aerofoil aerodynamics by a locally analytic method

AbstractA complete first‐order model and locally analytic solution method are developed to analyse the effects of mean flow incidence and aerofoil camber and thickness on the incompressible aerodynamics of an oscillating aerofoil. This method incorporates analytic solutions, with the discrete algebraic equations which represent the differential flow field equations obtained from analytic solutions in individual grid elements. The velocity potential is separated into steady and unsteady harmonic parts, with the unsteady potential further decomposed into circulatory and non‐circulatory components. These velocity potentials are individually described by Laplace equations. The steady velocity potential is independent of the unsteady flow field. However, the unsteady flow is coupled to the steady flow field through the boundary conditions on the oscillating aerofoil. A body‐fitted computational grid is then utilized. Solutions for both the steady and the coupled unsteady flow fields are obtained by a locally analytic numerical method in which locally analytic solutions in individual grid elements are determined. The complete flow field solution is obtained by assembling these locally analytic solutions. This model and solution method are shown to accurately predict the Theodorsen oscillating flat plate classical solution. Locally analytic solutions for a series of Joukowski aerofoils demonstrate the strong coupling between the aerofoil unsteady and steady flow fields, i.e. the strong dependence of the oscillating aerofoil aerodynamics on the steady flow effects of mean flow incidence angle and aerofoil camber and thickness.

The Lift On the Symmetrical Joukowski Aerofoil in a Stream bounded by a Plane Wall

In this paper the lift on the symmetrical Joukowski aerofoil of small thickness in the presence of a plane rigid wall is obtained in the form of a power-series, with the intention of discussing the manner in which the ground-effect on the lift of an aerofoil is modified by its thickness. The calculation of the lift is carried out by extending Green's analysis. It is found that in case when the symmetrical Joukowski aerofoil of small thickness is placed in the vicinity of the wall at small angles of attack as used in practice, the wall-effect on the lift is quite similar to that in the case of a plane aerofoil. Namely, as the symmetrical Joukowski aerofoil approaches the wall, the lift first decreases and then tends to increase to values which are greater than the corresponding values for the symmetrical Joukowski aerofoil in an infinite stream, especially when the angle of attack is sufficiently small. Further it is found that for any value of the distance of the aerofoil from the wall, the lift ratio Y / Y 0 decreases as the thickness increases even for small values of the angle of attack.

任意翼列の理論

In a previous paper, the present author has given a new method of treating potential flow past arbitrary aerofoils. In this paper, the same method is applied to arbitrary wing lat-tices. As an illustration. numerical calculations are carried out for a particular wing lattice composed of the Joukowski aerofoils. The results are satisfactory.

Book Reviews

Although this volume has only just come to hand it bears the date 1938 and it is, therefore, not surprising that most of the contents are considerably out of date. The book, which is written in French, begins by giving a general description of the wind tunnel at the Ecole Polytechnique at Bucarest. The general layout of the wind tunnel is very similar to that of the compressed air tunnel at the N.P.L. but the tunnel works with air at normal atmospheric pressure and the main structure is built of wood. The exterior casing of the tunnel is in effect an independent building exposed to the weather and takes the form of a horizontal octagonal prism with truncated pyramidal ends. The working section is 1·5 metres in diameter and a driving motor of 40 kw. gives a top speed of 46 metres per second or very roughly 150 ft. per second. Results of tests on a considerable variety of Joukowski aerofoils carried out at a Reynolds number of a little below half‐a‐million are given in the report. On account of this low Reynolds number and of the fact that the sections are not of types now used in practice, the data will be of little interest to the aircraft industry. One minor point of terminology which appears to be worthy of comment is that profit a didre means a profile having a finite trailing edge angle, the idea being that there is a dihedral angle between the tangent planes at the trailing edge.

On the Equation of Joukowski's Aerofoils

On the Equation of Joukowski's Aerofoils

The Flow of a Perfect Fluid through Cascades of Aerofoils

SummaryThe paper contains a generalisation of the conformal transformation by means of which a circle is converted into Joukowski aerofoils. The transformation is applied to a cascade of oval bodies which is the corresponding generalisation of the isolated circle; and it converts the oval bodies into a wide variety of cascades of curved shapes, which are generalisations of the isolated Joukowski aerofoils. In one particular case, the transformation yields a grid of flat laminae. Since the perfect fluid flow through such a grid is known (it appears to have been first given by Joukowski) the perfect fluid flow past the ovals, in any direction, and with equal but otherwise arbitrary circulation round each, is obtained. Accordingly, the perfect fluid flow through any of the derived cascades can be obtained.

The two-dimensional hydrodynamical theory of moving aerofoils—I

1—The two-dimensional hydrodynamical theory of aerofoils has hitherto been developed only from the standpoint of the aerofoil at rest in a moving liquid. In spite of the many disadvantages attaching to this method (no account can be taken in it of the accelerations or the angular velocity of the aerofoil), it seems to be the only one which up to the present has been found to be capable of development to any serious extent. In an important paper Wagner (1925) seems to have made the first real attempt to deal with accelerated aerofoils, but his discussion is limited to a flat plate in motion without rotation and in a constant direction to which it is infinitely slightly inclined. Later Glauert (1929) in a paper issued by the Aeronautical Research Committee, basing himself on the somewhat artificial concept of bound vortices, deals with a more general case, but his formulae still require a knowledge of the velocity potential of the fluid motion, which he was unable to give, except for the well-known case of the elliptic cylinder. Later, Lamb (1929) gave a revised form of Glauert’s discussion but was still unable to derive definite formulae for any specific case beyond those given by previous writers. In several papers published about this time Bickley (1929) succeeds in framing a more general mode of attack in these problems and pursues it to the extent of finding the force and couple on the cylinder in uniform motion—including rotation, applying his results in particular to the Joukowski aerofoils: but the method of attack employed by him is so cumbrous that it is difficult to develop to any extent in more complicated cases, and he does not take into account the accelerations of the motion in the general case. In undertaking a re-examination of two-dimensional potential theory in general, a method was discovered which enables the results for the more general theory to be developed with considerable ease, and my object in this paper is to sketch the application of the method to the general aerofoilshaped cylinder moving in any general manner in homogeneous incompressible fluid, including also the case when there is circulation round the cylinder.