Trailing-edge Conditions Research Articles

Direct Numerical Simulation of the Airfoil Segment's Flutter and its Effect on the Aerodynamic Force

This article presents numerical simulation of planar potential flow around an airfoil with possibility of changing its shape. Two-dimensional unsteady flow model with scalar velocity potential, which allows us to calculate pressure distribution along an airfoil from Cauchy-Lagrange integral, is used. For this purpose, an airfoil contour is approximated by a complex cubic spline with possibility of displacement its vertices. This algorithm has been used in the context of fluid-structure interaction and has been applied successfully to determination of stability of an elastic airfoil segment interacting with a flow stream, so-called panel flutter problem. Calculation of external flow is carried out by vortex panel method with Kutta-Joukowski trailing edge condition, which makes mathematical solution unique. Using this method of approximation of an airfoil in combination with the method of discrete vortices provides a semi-analytical solution for complex potential for whole computational domain of air flow. This solution significantly accelerates process of numerical computation of time-averaged aerodynamic force as well as the dynamic stability problem for aeroelastic wing design and temporal evolution of its natural disturbances.

Singularities in BIEs for the Laplace equation; Joukowski trailing-edge conjecture revisited

The paper deals with trailing-edge issues connected with the analysis of three-dimensional incompressible quasi-potential flows (i.e. flows that are potential everywhere, except for a zero-thickness vortex layer, called the wake). Specifically, following the Joukowski conjecture of smooth flow at the trailing edge, all the trailing-edge conditions that are required to avoid singularities in the boundary integral representation for the velocity, in a quasi-potential incompressible flow around a wing, are identified. In particular, these include the Kondrat'ev and Oleinik singularity as well as the vortex-line and the edge-jet singularities of Epton. Also, following Mangler and Smith, the behavior of the wake geometry at the trailing edge is determined, using the Kutta condition of no pressure discontinuity at the trailing edge. Specific theoretical issues are addressed which include (1) the relationship between Joukowski conjecture and Kutta condition, and (2) identification of those trailing-edge conditions that are necessary to assure the uniqueness of the solution (as opposite to relationships that are automatically satisfied by the solution). Regarding the first issue, in the main body of the paper, the Joukowski conjecture and the Kutta condition are used as if they were independent assumptions; then, in Appendix A, it is shown that the Kutta condition need not be invoked as a separate assumption since it may be obtained as a consequence of the governing equations and of the Joukowski conjecture. In order to clarify the second issue, the theoretical analysis is coupled with a numerical one. In particular, the conditions necessary to insure uniqueness are inferred (not proven) through numerical experimentation: only the no-vortex-line condition appears to be necessary to insure uniqueness. This is accomplished by using a piecewise-cubic boundary-element method for quasi-potential flows that is an extension of a high-order formulation introduced by the authors and their collaborators (the order of the formulation is adequate to address all the theoretical trailing-edge conditions uncovered). The emphasis is on steady flows in simply connected regions; however, some issues related to unsteady flows in multiply connected regions are also examined. Finally, several open problems that require additional work are identified.

Mechanical modeling of palatal snoring.

The mechanism of human snoring caused by vibration of the soft palate and the characteristics of the noise are investigated. The soft palate becomes unstable and vibrates violently once the inspiratory flow exceeds a critical speed. The physiological phenomenon is modeled by studying flow over a flexible plate. In determining the stability of this flow, the trailing edge conditions are crucial. It is found that the noise generated in the simple experimental configuration has distinct characteristics found in human snores. For example, there is an antiphase relation between unsteady pressures from the oral and nasal channels and this provides a feature that distinguishes snoring by vibration of the soft palate from that caused by other parts of the human airway.

Sound diffraction and dissipation at a sharp trailing edge in a supersonic flow

This paper is concerned with the process of energy conversion between acoustic waves and vorticity when plane sound waves impinge upon a sharp trailing edge of a semi-infinite plate positioned in a supersonic mean flow. It is shown that the diffracted field, confined to the Mach cone emanating from the trailing edge, is everywhere bounded and vanishes at the edge so that the issue of satisfying the Kutta condition at the sharp edge does not arise, which would require, in subsonic mean flows, a circulation to be imposed to the plate in order to remove the singularity at the edge, consequently resulting in vortex shedding from the plate. In the case of supersonic mean flows, no such circulation is needed because there is no singularity at the trailing edge, and the vorticity shed from the plate is entirely due to the discontinuity across the plate which is produced by the incident waves independently on the trailing edge conditions, and is swept downstream of the plate by the mean flow, forming a vortical wake behind it. The incident waves in this case completely determined the strength of the vortical wake as if the trailing edge did not exist, which is fundamentally different from the case of subsonic mean flows. It is shown, however, that energy exchange between sound and vorticity still occurs because, similarly to the case of subsonic mean flows, the vorticity, once shed into the sound field, experiences a lift force that does work as it is swept downstream by the mean flow. This process may either dissipate or produce acoustic energy, depending on the angle of the incident waves and the Mach number of the supersonic mean flow.

Helmholtz decomposition revisited: Vorticity generation and trailing edge condition

The use of the Helmholtz decomposition for exterior incompressible viscous flows is examined, with special emphasis on the issue of the boundary conditions for the vorticity. The problem is addressed by using the decomposition for the infinite space; that is, by using a representation for the velocity that is valid for both the fluid region and the region inside the boundary surface. The motion of the boundary is described as the limiting case of a sequence of impulsive accelerations. It is shown that at each instant of velocity discontinuity, vorticity is generated by the boundary condition on the normal component of the velocity, for both inviscid and viscous flows. In viscous flows, the vorticity is then diffused into the surroundings: this yields that the no-slip conditions are thus automatically satisfied (since the presence of a vortex layer on the surface is required to obtain a velocity slip at the boundary). This result is then used to show that in order for the solution to the Euler equations to be the limit of the solution to the Navier-Stokes equations, a trailing-edge condition (that the vortices be shed as soon as they are formed) must be satisfied. The use of the results for a computational scheme is also discussed. Finally, Lighthill's transpiration velocity is interpreted in terms of Helmholtz decomposition, and extended to unsteady compressible flows.

Possibility of Various Airfoil Shape Modes and Their Steady State Stability of Single Membrane Sailwing.

In order to estimate various modes of sailwing aerofoil shapes (a number of such modes can exist) and their steady-state stability in the single-membrane sailwing, a two-dimensional method of analysis is presented for trailing-edge conditions in which the membrane tension constant, the membrane slackness constant, and the elasticities of the trailing-edge wire and membrane are considered systematically. Calculation examples are given to show the various modes of aerofoil shapes and their steady-state stability for each trailing-edge condition. The development of the analysis method, and the calculation examples, are presented in some detail and discussed, and several conclusions are drawn.

Linearized planing-surface theory with surface tension. Part II: Detachment with discontinuous slope

AbstractIn Part I of this series, surface tension was included in the classical two-dimensional planing-surface problem, and the usual smooth-detachment trailing-edge condition enforced. However, the results exhibited a paradox, in that the classical results were not approached in the limit as the surface tension approached zero. This paradox is resolved here by abandoning the smooth-detachment condition, that is, by allowing a jump discontinuity in slope between the planing surface and the free surface at the trailing edge. A unique solution is obtainable for any input planing surface at fixed wetted length if one allows such jumps at both leading and trailing edges. If, as is the case in practice, the wetted length is allowed to vary, uniqueness may be restored by requiring either, but not both, of these slope discontinuities to vanish. The results of Part I correspond to the seemingly more-natural choice of making the trailing-edge detachment continuous, but it appears that the correct choice is to require the leading-edge attachment to be continuous.

Sound diffraction at a trailing edge

The diffraction of externally generated sound in a uniformly moving flow at the trailing edge of a semi-infinite flat plate is studied. In particular, the coupling of the sound field to the hydrodynamic field by way of vortex shedding from the edge is considered in detail, both in inviscid and in viscous flow.In the inviscid model the (two-dimensional) diffracted fields of a cylindrical pulse wave, a plane harmonic wave and a plane pulse wave are calculated. The viscous proess of vortex shedding is represented by an appropriate trailing-edge condition. Two specific cases are compared, in one of which the full Kutta condition is applied, and in the other no vortex shedding is permitted. The results show good agreement with Heavens’ (1978) observations from his schlieren photographs, and confirm his conclusions. It is further demonstrated, by an explicit expression, that the sound power absorbed by the wake may be positive or negative, depending on Mach number and source position. So the process of vortex shedding does not necessarily imply an attenuation of the sound.In the viscous model a high-Reynolds-number approximation is constructed, based on a triple-deck boundary-layer structure, matching the harmonic plane wave outer solution to a known incompressible inner solution near the edge, to obtain the viscous correction to the Kutta condition.

Trailing Edge Conditions for Unsteady Flows at High Reduced Frequency

In the prediction of time-variant flows past isolated airfoils and airfoil cascades, the steady-state KuttaJoukowsky condition is generally assumed to be valid. Recent experimental investigations on isolated airfoils however, have indicated that this assumption is only appropriate for flows characterized by low reduced frequency values. For the case of airfoil cascades, only low reduced frequency data exist, but these limited data indicate the same trend, i.e., the Kutta-Joukowsky condition is appropriate only for low reduced frequency values. Hence, the objective of the experimental investigation described herein was to determine the applicability of the Kutta-Joukowsky condition for isolated airfoils and airfoil cascades at high reduced frequency values over a range of incidence angles. This was accomplished by measuring the time-variant surface pressure distribution in the trailing edge region of a classical isolated flat plate, a classical flat plate cascade, and a cambered airfoil cascade, and correlating these data with an appropriate state-of-the-art zero incidence flat plate theoretical model that applies the Kutta-Joukowsky condition. The results obtained indicate that at these high reduced frequency values, the Kutta-Joukowsky condition is appropriate for the classical isolated flat plate and flat plate cascade over a wide range of incidence angles. However, for the cambered airfoil cascade, this condition does not appear to be satisfied for any value of incidence angle.

Analysis Method for Viscous Flow over Circulation-Controlled Airfoils

A method developed for the analysis of the incompressible viscous flow over circulation-controlled airfoils is described. A surface vorticity method is used to solve the inviscid portion of the flow, and a combination of integral and finite-difference methods is used to calculate the development of the viscous layers. An iterative process is used to arrive at final solutions which satisfy an appropriate trailing-edge condition and incorporate the interaction between the viscous and potential regions of the flow. Comparisons between calculated and experimental results show good agreement for surface pressure distributions and lift coefficients over a range of blowing momentum coefficients from 0 to 0.12. Nomenclature c = airfoil chord length CL =lift coefficient, based on airfoil chord length Cp = static pressure coefficient CM = blowing momentum coefficient = // l/2p U^ c H ^ = boundary-layer shape factor = b*/Q J = blowing momentum flux per unit span 7ex = excess momentum flux in wall jet per unit span / = length of panels used to represent airfoil N = number of panels used to represent airfoil p = static pressure q = surface source strength (volume efflux per unit surface area) R = radius of curvature Re = momentum thickness Reynolds number = U6/v s = arc length along airfoil surface u, v = flow velocity components in potential flow analysis U = local flow velocity at outer edge of viscous layer C/oo = freestream velocity 7 = strength per unit area of surface vorticity distribution = circulation around airfoil = boundary-layer displacement thickness = boundary-layer momentum-defect thickness = fluid viscosity = fluid density

The 1976 Freeman Scholar Lecture: Some Current Research in Unsteady Fluid Dynamics

Important unsteady fluid dynamic effects occur in a wide range of modern engineering problems. A review and critical appraisal has been made of the current research activities on topics that contain essential and unique unsteady features, especially those which cannot be approximated by quasi-steady analyses. A synopsis of the main areas covered in this paper is given below. Linear potential theory is well advanced and most of the fundamental concepts are well understood. The theory has been specially adapted for engineering purposes to many complex geometries and flow environments, but its limitations are not well established in most cases. Transonic flows have received considerable attention in recent years, and the profusion of numerical analyses of nonlinear unsteady flows has outstripped measurements. However, new experimental investigations are underway. Numerical codes are becoming much more efficient, and efforts are being made to incorporate viscous effects into them. Unsteady boundary layers have been computed with almost no complementary experimental guidance, and this deficiency is particularly acute in the turbulent case. A major conceptual difference between steady and unsteady separation has been identified and is continuing to be studied. Unsteady stall is currently under detailed examination, and recent experiments have shed considerable new insight on the fundamental mechanisms of dynamic stall on oscillating airfoils. New attempts to treat unsteady stall as a strong viscous-inviscid interaction problem are needed. Vortex shedding from bluff bodies is difficult to predict, especially in cases where body oscillations are self-induced by the fluctuating fluid dynamic forces. Nonlinear oscillator models are limited by a lack of understanding of the basic fluid dynamic phenomena. The trailing edge condition of Kutta and Joukowski for thin airfoils has been called into question recently for unsteady flows at high frequencies or with trailing-edge separation. The correct modeling of this condition is important in predicting the fluid dynamic forces on all thin lifting surfaces that fluctuate. Considerable progress has been made in each of these subjects, but none of them has been mastered. The questions that remain unanswered pose intriguing challenges to the fluid dynamics community.

Über eine physikalische begründung der kuttaschen abflussbedingung für die stationäre strömung eines idealen fluids um ein traglügelprofil

A new foundation for Kutta's trailing edge condition is given for an almost perfect fluid flow by means of the principle of least universal dissipation. Formation of vortices appears here as a means of the nature to minimize dissipation level.

On the remarkable accuracy of the vortex lattice method

Integral equations with Cauchy type singular kernels are of frequent occurrence in aerospace engineering and usually require solution by complicated numerical techniques because of singularities in the solution itself. The cases of steady or unsteady planar subsonic wing theory are typical and recently a simple, but remarkably accurate, method known as “Vortex Lattice” has been found very satisfactory. In this paper a simplified analog (the finite Hilbert transform) or the equation of two-dimensional incompressible thin wing flow, is studied and several important points concerning this discretization proved. For instance, it is unique and does not require external imposition of trailing-edge conditions; the singularities in the solution are automatically generated without the use of special loading functions —provided a specific and unique spacing for the collocation points is adhered to; the accuracy is of second order for all bounded input functions, and the behavior of the system for different collocation points smoothly generates different but useful singular solutions.

Correlation of Total Lift Data for Thin, Sharp-Edged, Low-Aspect-Ratio Delta Wings at Low Speeds

The amount of research carried out in the field of highly swept wings has reached massive proportions. No attempt is made to review the literature on this subject but it seems that despite all efforts, no method yet exists for calculating the lift of delta wings in subsonic flow where the trailing-edge condition makes true conical flow impossible. A recent paper by Nangia and Hancock, has attempted this very difficult problem but the method proposed is not yet suitable for general application. Greater success has been obtained in supersonic flow,see for example ref.2.Various formulae have been obtained for the lift coefficient of delta wings in conical flow and the lift can be divided into two parts; the linear part where the flow is fully attached in the region of zero incidence and the nonlinear part where extra lift is obtained from the leading edge vortices. One typical formula is derived from a relatively simple theory by Edwards.

Further Discussion of Broad-Band Fan Noise

This paper continues the discussion of broad-band fan noise concepts begun in a paper presented to the November 1970 ASA meeting [J. C. Kelly, “Broad-Band Fan Noise,” J. Acoust. Soc. Amer. 49, 108–109(A) (1971)]. It reopens by questioning the total validity of unsteady aerofoil theory. In particular, it examines the importance of the trailing edge condition both from an aerodynamic and an acoustic viewpoint. It also asks whether the modeling of flow distortions as two-dimensional gusts can be upheld at the higher reduced frequencies. The conclusion is that it is probably necessary to discard such theory altogether for the high reduced frequency engine flow distortion fan interactions which we expect to generate broad-band noise. The more positive aspect of this second paper is the substitution of a model that can discern between propagating and nonpropagating reduced frequency components of the flow over a finite surface. This leads to a delineation between M8, M6, and M3 power law regimes for acoustic radiation from the surface.

Aerodynamic Loading Induced on a Two-Dimensional Wing by a Free Vortex in Incompressible Flow

Practical problems often arise in which the aerodynamic loadings on wings and fins, due to nearby free vortices, are required. For example, aerodynamic loads on helicopter blades are affected by the tip vortices which are continuously being shed and which lie beneath the rotor plane. And vortices shed from the nose of an aircraft fuselage can induce significant side loads on a fin.The estimation of the aerodynamic load distribution on a moving wing in the neighbourhood of a free vortex (whose axis is primarily in the free stream direction) is an interesting problem which, as far as the author is aware, has not been discussed previously in the literature. Standard techniques for conventional linearised wing theory cannot be applied directly to this problem because the trailing vorticity shed from the wing trailing edge is affected by the flow field of the free vortex and so the mathematical application of the Kutta trailing edge condition is more complex.

Theory for Finite-Width High-Speed Self-Acting Gas-Lubricated Slider (and Partial-Arc) Bearings

The behavior of the pressure distribution within partial-arc and slider bearings under conditions of high-speed operation is investigated to provide an understanding of the basic phenomenon. Both the trailing-edge conditions and side-leakage effects are treated by asymptotic methods. A design example is given in which the edge effects on load capacity are computed as “corrections” to the infinite speed (Λ → ∞) solution.

Establishment of Lift on an Aerofoil with a Jet Flap: A Theoretical Note Relating to a Wing with Trailing Edge Jet with Extension to Cover Wings of Finite Span

The circulation which is established around an aerofoil section which has i trailing edge jet, and thus also the lift, is directly dependent on a trailing edge condition similar to the Joukowski condition in conventional steady and unsteady flow. There is a downwash made up of elementary vortices but in this case the vortex density is not zero when the sheet extends to infinity. The problem is govcrnec by an equation similar to the Wagner equation, into which is introduced the curvature of the downwash at the trailing edge. Certain precautions have to be taker when calculating this. The law governing the movement of the elementary vortices in the downwash plays a very important part. It is directly dependent on the viscous damping of the vorticity. This effect has also had to be taken into consideration, and the drag law of the jet behind the nozzle established. Other results follow concerning the solution of the transient case, and the determination of the circulation up to the moment of reaching its steady limit value, which is achieved in finite time. The velocity distribution on the aerofoil can be found at any instant, and thus also all the properties concerning the aerodynamic forces. The method has been applied to two cases, and a comparison is made with experimental results. An extension to the case of a conventional aerofoil (without jet) is possible; in this case the circulation must have a slightly lower value than in the Joukowski case, the difference depending on the Reynolds number. There is a potential drag resulting from the existence of the downwash and its loss of momentum. In Part II the finite span case is considered. The basic equations are established, and an approximate solution of the steady How case found, where the distribution of circulation is elliptical. A wing interrupted by a fuselage is also considered, and comparison with experimental results given.