Airfoil Problem Research Articles

Variational principles for the transonic airfoil problem

The transonic airfoil problem is formulated, and associated variational principles are constructed. Both the full potential form of the governing equations and small disturbance theory are considered. Variational functional are developed for the full potential equation using a modified Plateau integral and also directly from the continuity equation. A simpler nonlinear functional is obtained for the small disturbance approximation by direct divergence manipulations to a weak solution form of the Galerkin requirement. For slightly supercritical flows, iterative methods and Mach number perturbation techniques lead to simpler but less general variational principles. Appropriate finite element formulations are developed using the principles.

Oscillatory Subsonic Piecewise Continuous Kernel Function Method

Theme T WO main methods currently are employed in the prediction of the aerodynamic forces acting on lifting surfaces at subsonic flow. These methods inlcude the subsonic kernel function approach, which assumes pressure polynomials to describe the pressure field over the wing, and the finite-element approach, such as the doublet lattice method (DLM) or the vortex lattice method (VLM). The kernel function method (KFM), when based on orthogonal polynomials and carefully determined collocation points, shows a rapid convergence of its solution (with a small number of pressure polynomials) provided that the pressure field over the wing is smooth. Pressure discontinuities, such as those arising from control surface rotations, can be treated successfully using the KFM only when the exact shape of the singularity is known. The overlooking of pressure singularities using the KFM leads to a rapid deterioration in the convergence of the solution and to a general loss in the effectiveness of the method. The finite-element methods (FEM) can cope successfully with unknown pressure singularities provided that their location is known. The FEM, however, require a relatively large number of unknowns for convergence, leading at times to a relatively large residual error at the converged values. In the present paper a method is presented, similar to the one described in Ref. 4 in connection with mixed transonic flow, which combines the rapid convergence characteristics of the KFM with the ability of the FEM to treat discontinuities without having to determine their exact form. The method is tested using a two-dimensional airfoil problem (with control surfaces and gaps) with the intention of establishing its merits before embarking on its extension to the three-dimensional flow case.

Convergence Acceleration of Relaxation Solutions for Transonic Flow Computations Mohamed

The problem of how to speed up the convergence of currently available iterative methods for transonic flow computations with minimal alterations in computer programing and storage is considered. A cyclic iterative procedure applying nonlinear sequence transformations akin to those of Aitken and Shanks is developed. Based on the power method, the errors in these sequence transformations are studied. Examples testing the procedure for model Dirichlet problems and for transonic thin airfoil problems show that savings in computer time of a factor of two to five, or more, is generally possible, depending on accuracy requirements and the particular iterative procedure used. I. Introduction M ANY current methods of fluid dynamic computations make use of relaxation procedures. There are several aspects of the computation that considerably limit the usefulness and potentiality of these programs. One is the slow convergence with respect to iterations of the flowfield calculation, and, hence, costly computer time. This paper presents studies on how to speed up the convergence of currently available iterative procedures with minimal alterations in computer programing and storage requirements. To see the need of the convergence acceleration, we may take as an example the finite-difference solution to Dirichlet's problem on a unit square. The convergence rate of practical iterative procedures (Jacob!, Gauss-Seidel, Successive Over

Low-frequency approximations in unsteady small perturbation subsonic flows

A more rigorous proof is given of the validity of a generalized Prandtl—Glauert technique for caculating the solution to time-dependent small perturbation flows. The method, first used by Miles (1950a) for the airfoil problem and later applied by Amiet & Sears (1970) to more general problems, allows calculation of the term of first order in frequency. Anomalous behaviour for the two-dimensional problem is examined in detail and found to be limited to those two-dimensional cases which include shed vorticity downstream of the body. This anomaly, which precludes using the method for these cases, results from the need to satisfy a velocity boundary condition on the body. For this purpose the velocity must be calculated from the basic variable, the pressure, through an integrated form of the momentum equation. It is in thus calculating the velocity that the anomaly occurs. The method can be applied to both the two-dimensional case without shed vorticity and the general three-dimensional case.

An Improved Solution of the Two-Dimensional Jet-Flapped Airfoil Problem

3 Partridge, D. W. and Pecover, B. E., An Application of the RAE Wind-Tunnel/Flight Dynamics Simulator to the Speed Dynamics of a Slender Delta Aircraft (HP 115), Repts. and Memos. 3669, 1969, Aeronautical Research Council, London, England. 4 Engler, P. B. E. and Moss, G. F., Low-Speed Tunnel Tests on a ith Scale Model of the Handley Page HP 115, Repts. and Memos. 3486, 1965, Aeronautical Research Council, London, England. 5 Thompson, J. S. and Fail, R. A., Derivative Measurement on Sting-Mounted Wind-Tunnel Models at RAE Bedford, AGARD Current Paper 17, 1966. 6 Beecham, L. J. and Titchener, I. M., Some Notes on an Approximate Solution for the Free Oscillation Characteristics of Nonlinear Systems Typified by x + F(x,x) = 0, Repts. and Memos. 3651, 1969, Aeronautical Research Council, London, England. 7 KrylofT, N. and Bogoliuboff, N., Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, N.J., 1937. 8 Ross, A. J. and Beecham, L. J., An Approximate Analysis of the Nonlinear Lateral Motion of a Slender Aircraft (HP 115) at Speeds, Repts. and Memos. 3674, 1971, Aeronautical Research Council, London, England. 9 Henderson, J. M., Low Speed Handling of a Slender Delta (HP 115), Journal of the Royal Aeronautical Society, Vol. 69, No. 653, 1965, pp. 311-324. 10 Thompson, J. S., Fail, R. A., and Inglesby, J. V., Low Speed Wind-Tunnel Measurements of the Oscillatory Lateral Stability Derivatives for a Model of the Slender Aircraft (HP 115) Including the Effects of Frequency Parameter, Current Paper 1097, 1969, Aeronautical Research Council, London, England.

Comments on the Two Notes "Correct Formulation of Airfoil Problems in Magnetoaerodynamics"

Comments on the Two Notes "Correct Formulation of Airfoil Problems in Magnetoaerodynamics"

Correct formulation of airfoil problems in magnetoaerodynamics

Correct formulation of airfoil problems in magnetoaerodynamics

Comment on "Numerical Lifting Surface Theory-Problems and Progress"

N a recent paper, Landahl and Stark1 present a progress report on the status of numerical approaches to nonsteady lifting-surface theory for planar and nonplanar configurations as applied to the linearized thin-wing problem, with particular stress on the subsonic case. Over the past few years, in the course of studies adapting unsteady lifting-surface theory to marine propellers,24 Davidson Laboratory has developed a new method for the solution of the downwash surface integral equation. By proper expansion of the kernel function and introduction of the so-called lift operator/7 the chordwise integration is performed analytically with the additional advantage that the numerical solution is greatly simplified. These studies indicate that use of the generalized lift operator, which is in fact dictated by the nature of the integral equation itself, is a more accurate and rapid procedure than the usual numerical approaches for evaluating the steady and unsteady pressure distributions on lifting surfaces and resultant hydrodynamic forces. This technique has been used in Ref. 5, where the lifting surfaces are the blades of a marine propeller operating in nonuniform inflow, and in Ref. 6 for the case of a deeply submerged, flat, rectangular hydrofoil in steady flow. In Ref. 7, this new approach has been applied to several two-dimensional, unsteady airfoil problems and has yielded results identical to the known explicit solutions.

A New Procedure for the Solution of Lifting-Surface Problems

A new method for determining the steady and unsteady pressure distributions on lifting surfaces is presented. The method employs a technique which is found to be a more accurate, versatile, and rapid procedure (requiring less computer time) for inverting the downwash integral equation than the presently used mode-collocation method. The generalized lift-operator technique allows the chordwise integration to be carried out analytically, thereby avoiding the difficulties in the presently used method. This new approach has been applied to several two-dimensional, unsteady airfoil problems to demonstrate compatibility with known explicit solutions. Numerical solutions for two rectangular foils of aspect ratio 1 and 2 are also presented. The values show good agreement with experiment and fit in with known trends.

Correct formulation of airfoil problems in magnetoaerodynamics.

Correct formulation of airfoil problems in magnetoaerodynamics.

Solution of the transonic airfoil problem through parametric differentiation.

The difficulties associated with the nonlinearity of the transonic small disturbance equation are avoided by considering the linear problem governing the rate of change of the flow velocity with respect to the airfoil thickness ratio parameter and integrating solutions of this problem over the thickness ratio. This procedure is equivalent to summing a series of perturbations in airfoil thickness and has the advantage of moving all the nonlinearity to a firstorder ordinary differential equation, where it causes little difficulty. Solutions for nonlifting airfoils exhibit all the commonly observed features of transonic flow, such as the formation and movement of a shock on the airfoil surface and the presence of a detached shock upstream of the airfoil for low supersonic Mach numbers. Numerical results are given for subsonic and supersonic flow and for flows at Mach number unity which are in general agreement with other theoretical results and with experiment. Preliminary results for mixed flows with subsonic freestream Mach numbers indicate a rearward movement of the shock impinging on the airfoil surface with increasing airfoil thickness and exhibit the commonly observed rapid acceleration behind the shock. Extensions to lifting and unsteady flows are also discussed, as well as possible applications of the method to other problems in fluid mechanics.

An Improved Computational Model for Predicting the Unsteady Aerodynamic Loads of Rotor Blades

The computation of the time‐varying air loads of rotor blades is considered with the view of improving the treatment of unsteady aerodynamic effects in currently available programs without unduly increasing the required computational time. A computational model is proposed for accomplishing this objective and is evaluated by applying it to the two‐dimensional oscillating airfoil problem for which the exact solution is available.

An improved glauert series for certain airfoil problems

An improved glauert series for certain airfoil problems

Constant shear flow past an airfoil with arbitrary surface suction

The problem of two-dimensional airfoils of any shape placed in constant shear flow when either tangential or normal velocity is prescribed on the airfoil surface, has been studied. The complex variable method is employed to find the stream-function and the forces on the airfoil. Cases of over-all suction localised and multi-slot suction are considered. In a numerical example on over-all suction the lift and drag forces are evaluated. It is concluded that the case of shear flow gives promise of considerable lift increase over the potential flow case, though there is a corresponding increase in drag also.

An Approximate Solution of the Lifting Problem of Thin Airfoils at Sonic Speed

An Approximate Solution of the Lifting Problem of Thin Airfoils at Sonic Speed

Loads on a Supersonic Wing Striking a Sharp-Edged Gust

This is a calculation of the chordwise lift distribution, total lift, and moment on a two-dimensional wing striking a sharp-edged gust a t supersonic speed. A direct solution is established by considering a distribution of sources in a fluid at rest. An alternate method using Busemann's conical flow is also shown to be applicable. The time history of the total lift and mid-chord moment is discussed. I t is shown tha t the total lift increases with time and reaches a maximum that corresponds to the steadystate phase of the flow. The mid-chord moment goes through a maximum independent of the Mach Number if the latter value is larger than 4/7r, while this maximum can become infinite for a range of Mach Numbers between 4/TT and 1. (1) INTRODUCTION CONSIDERABLE ATTENTION has been given lately to nonstationary flow problems of wings flying at supersonic speeds. Most of the work, however, has been concerned with the aerodynamic forces on an oscillating airfoil from the standpoint of flutter analysis. The problem of the wing hitting a sharp-edged gust is of a different nature and turns out to be actually much simpler than the oscillating airfoil problems. It is shown in section 2 that it may be treated by a distribution of sources of a simple type along the chord and that the pressure distribution may be derived by elementary methods. The procedure does not introduce a moving fluid but considers a fluid at rest in which nonstationary sources are distributed in a layer of variable extent. This point of view, which is closer to acoustics than to aerodynamics, is somewhat novel and seems to present advantages of simplicity and closeness to physical reality in certain categories of problems. The pressure distribution derived by this method is applied to the calculation of the time history of lift and moment on the wing in section 3. Particular attention is given to the value of the mid-chord moment;.-which starts from zero, rises to a maximum,,and goes back to zero. The value of this maximum and related 4ata is evaluated in section 4. These results are of particular interest to the designer. The derivation of the pressure as given in section 1 is only one of the methods that may be used in this problem. As an independent check and as an illustration of the application of Busemann's method of conical flow to a nonstationary problem, an alternate derivation is given for the pressure distribution in section 4. Received August 15, 1948. * Member of Cornell Aeronautical Laboratory consulting staff. Also Professor of Applied Physical Sciences, Brown University. In a paper by Schwarz procedures used in oscillating airfoil theory are extended to the problem of a wing striking a sharp-edged gust at supersonic speed. Results for an oscillating down-wash lead to the gust problem by a Fourier integral representation. This method constitutes a considerable detour and introduces intermediate results of a transcendental nature which are actually not needed and are more complicated than the result. I t may be verified that the expression derived in the present paper for the pressure distribution is equivalent to that derived by Schwarz. He does not, however, discuss the physical aspects of the problem or derive expressions for lift and moment. (2) DERIVATION OF THE PRESSURE DISTRIBUTION The wing of chord / enters a uniform gust of upward velocity i/0 at the supersonic velocity V (Fig. 1). The velocity component normal to the wing must remain zero, and this condition is equivalent to the generation of a velocity normal to the wing which cancels the gust velocity (Fig. 2). This may also be considered as a reflection of the gust on the wing. Because the velocity is supersonic, the pressure distribution on. one side does not influence the pressure on the other, and therefore we need only consider the bottom side. The pressure distribution on top will be the same except for a reversal of sign. For the same reason the pressure distribution is not influenced by the trailing edge, and

A Simplified Two-Dimensional Theory of Thin Airfoils

IN THE EXISTING SOLUTIONS of the thin airfoil problem, the determination of the steady state lift and moment of an airfoil of small camber and thickness, as given by Birnbaum, von Karman, C. B. Millikan, etc., 4> 5 the velocity potential for the flow is determined, and the lift distribution and the integrated lift and moment are found from this result. It appears that a somewhat simpler solution of this problem can be obtained by working directly with the velocity components rather than with the velocity potential. This same method can also be used to advantage in many threedimensional wing problems. An important feature of this method is that the inverse problem, the determination of an airfoil shape that will produce a given lift distribution, can be solved just as easily as the direct problem. Both the direct and the inverse problem will be investigated by this method in the present paper. The notations used in the analysis are as follows:

Some Simplified Methods in Airfoil Theory

T USE OF the acceleration potential in thin airfoil theory was introduced by L. Prandtl.'' The method has since been applied by several authors' to problems of stationary and non-stationary flow in both compressible and incompressible fluids. A striking feature of this method is that no use has to be made of vortex wakes because the acceleration potential is everywhere continuous in the fluid. Such concepts as the trailing vortices and the induced downwash play no direct part in this theory. The purpose of the present article is the treatment of two dimensional airfoil theory in an incompressible fluid by the combined use of conformal transformation and the acceleration potential. This method of approach greatly simplifies the solution of certain airfoil problems. First the reader is introduced to the fundamental properties of the acceleration potential in an incompressible fluid. Then it is shown how the problem of finding the acceleration potential can be solved by conformal representation of the airfoil on a circle and a simple relation is derived between the velocity and the stream corresponding to this potential. The method is next applied to the stationary airfoil with and without flap, and to the determination of the airfoil camber line and thickness function for a given pressure distribution along the chord. The simplicity of the method is especially apparent for the case of the oscillating airfoil which is the subject of the last treatment. For the sake of brevity only translatory oscillations are considered.