Trefftz Plane Research Articles

Wake Forces Implied in the Theodorsen and Goldstein Theories of Propellers

This study examines wake edge forces and normal forces implied, but overlooked, respectively, in the Theodorsen and Goldstein theories of the optimum efe ciency of propellers. It is argued that the postulated trailing vortex sheets must coincide with real rigid surfaces to support forces exerted by the e uid. In the rigidized Theodorsen model, edge suction forces are tilted backward and circumferentially in the region of contraction. These are implicitly included in the predictions of propeller thrust and torque. The corrections are estimated herein. The resulting change in efe ciency, a gain, is normally less than 1%, but can reach over 2% in extreme examples. In the rigidized Goldstein model, the wake does not contract, but the streamlines do. The crosse ow component, acting on the bound trailing vortices, generates lift; hence, thrust and torque components. These corrections to the Goldstein-based predictions (using Theodorsen’ s formulas ) are found to be compatible with the contraction correction of the Theodorsen theory: the Goldstein theory, when rigidized, appears equivalent to the Theodorsen theory. In the Theodorsen theory, the required wake forces are seen to be so small that their neglect (by neglect of wake rigidity ) is of conceptual signie cance only. Nomenclature A = cross-sectional area of one turn of helicoid wake at z = ` projected on Trefftz plane, 2 pR `

FORCES AND MOMENTS IN INCOMPRESSIBLE QUASI-POTENTIAL FLOWS

An exact formulation, based on the Trefftz-plane approach, for the evaluation of aerodynamic loads acting on bodies having arbitrary motion, is presented for incompressible flows that are quasi-potential (i.e., potential everywhere except for the points of a zero-thickness wake surface). The resulting expressions, obtained without using simplifying assumptions, give an exact representation for forces and moments in terms of (i) the velocity potential over the body surface, (ii) the velocity potential and its normal derivative over an arbitrary plane that intersects the wake (Trefftz plane), and (iii) the potential discontinuity over the portion of the wake limited downstream by such a plane. It is shown that the present formulation includes, as particular cases, some classical results by Prandtl, Trefftz and Lamb, for the evaluation of forces and moments in incompressible quasi-potential flows. A convergence analysis is presented in order to assess the accuracy of the present formulation in comparison with the evaluation of forces and moments by direct integration of the pressure distribution over the body surface. Numerical results concerning lift, drag and moments acting on isolated wings in steady and unsteady flows are included.

The Ducted Tip—A Hydrofoil Tip Geometry With Superior Cavitation Performance

A novel hydrofoil design, consisting of a small diameter flow-through duct affixed to the tip, has been studied. The tip vortex cavitation inception index, σi, of this hydrofoil geometry is about a factor of 2 lower than that of a conventional rounded hydrofoil tip. This inception improvement comes with little associated performance penalty. For angles of attack greater than 8 deg the noncavitating lift-drag ratio is actually superior to that of an unducted hydrofoil of equal span, although with lower wing loadings the hydrofoil performance is diminished by application of the ducted tip. The ducted tip is effective at reducing the tip vortex inception index because, in contrast with the rounded tip, for which vorticity in the Trefftz plane is confined to a line, the ducted tip shed vorticity at the trailing edge is distributed over a line and circle. Distributing the vorticity in this fashion causes the trailing vortex to roll up less tightly, and hence have a higher core pressure and lower σi, than a conventional hydrofoil tip. It is also suspected that the interaction at the microscale level between the flow through the duct, and the flow around it, makes the vortex core size larger, and therefore σi smaller. The ducted tip design has many potential marine applications, including to ship and submarine propellers, submarine control fins, and ship rudders.

Computation of induced drag for elliptical and crescent-shaped wings

Recent interest in the induced drag characteristics of crescent-shaped wings has led to a closer look at the methods used for determination of induced drag from computational aerodynamic methods. Induced drag may be computed by integration of surface pressure or by evaluation of a contour integral in the Trefftz plane. A high-order panel method was used to study the induced drag of elliptical and crescent-shaped wings using both techniques. Induced drag computations using surface pressure integration were strongly affected by panel density and angle of attack. Drag computations for the crescent wing were especially sensitive to spanwise panel density because of the complex flowfield near the tip. Trefftz-plane results for the two wing planforms were not sensitive to panel density or angle of attack. The effect of correctly modeling the force-free, rolled-up wake geometry on the predicted span efficiency was demonstrated for both wing planforms. The span efficiency predicted from Trefftz-plane integration was about 0.97 for the elliptical wing and 0.99 for the crescent wing, both somewhat less than the classical theoretical maximum for planar wings. Most of the apparent drag reduction of the crescent wing claimed in previous studies was probably an artifact of the surface-pressure integration. The slightly higher span efficiency for the crescent wing was attributed to a more nearly elliptical spanwise lift distribution.

Errata: Asymptotic Methods for the Prediction of Transonic Wind-Tunnel Wall Interference

Errata: Asymptotic Methods for the Prediction of Transonic Wind-Tunnel Wall Interference

Asymptotic methods for the prediction of transonic wind-tunnel wall interference

Two limiting cases related to transonic wind-tunnel wall interference are considered. The first concerning slender airplanes is briefly treated only as it pertains to the main discussion relating to high aspect ratio configurations. Principal features of the slender airplane case are a three-deck flow structure as well as an area rule for interference. Pressure-specified wall conditions are incorporated by addition of averages and Fourier transforms of the wall pressures that modify the far field that controls the interference. For a high aspect ratio two-deck second case, the interference is driven by the imaging in the Trefftz plane projection of the experimental pressure-specified control surface of the (outer deck) trailing vortex system in the Trefftz plane

Hybrid trefftz plane elasticity elements with p ‐method capabilities

AbstractA family of p‐method plane elasticity elements is derived based on the hybrid Trefftz formulation.1 Exact solutions of the Lamé‐Navier equations are used for the intra‐element displacement field together with an independent displacement frame function field along the element boundary. The final unknowns are the parameters of the frame function field consisting of the usual degrees of freedom at corner nodes and an optional number of hierarchic degrees of freedom associated with the mid‐side nodes. Since the element matrices do not involve integration over the element area, the elements have a polygonal contour with an optional number of curved sides. The quadrilateral element has the same external appearance as the conventional p‐method plane elasticity element.2,3 But unlike in the conventional p‐method approach, suitable special‐purpose Trefftz functions are generally used to handle the singularity and/or stress concentration problems rather than a local mesh refinement. The practical efficiency of the new elements is assessed through a series of examples.

円形開風洞中の翼の最小誘導抵抗

The aerodynamic characteristics of a straight wing lying on a diameter in a circular open wind tunnel are studied, where minimum induced drag is assumed. In my previous paper, exact expressions of spanwise optimum lift distribution and minimum induced drag have been obtained from Trefftz plane flow field analysis, where an independent variable transformation and Sohngen's inversion integral formula were used. In this study, an alternative analysis is done, where a comformal mapping and Sohngen's formula are used. The results obtained here are coincident with those of Ref. 8.

The minimum drag of thin wings at supersonic speed according to Kogan's theory

In the theory of thin lifting surfaces the minimum drag consistent with a given total lift occurs when the downwash, averaged between forward and reversed motion of the wing, has the same value at all points of the wing planform. In Kogan's theory the conditions for minimum drag are determined on the forward sloping characteristic surface touching the trailing edge of the wing and it is shown that such a surface plays the role of the Trefftz plane familiar in subsonic wing theory. This paper shows how Kogan's theory may be applied to determine the drag of elliptic wings at supersonic speed. It appears that such wings have lower drag than the conventional “delta” wing.

Effect of downwash on the induced drag of canard-wing combinations

P o = Fourier coefficient in expansion of WT - Fourier coefficient in expansion of K2 = induced drag = biplane gap = circulation = elliptic circulation = induced circulation = lift the lift distribution on each surface is elliptical under the influence of the downwash field of the other, and it is clear that, in order to satisfy such a condition, a lift-dependent twist needs to be imparted to one or both surfaces. In this Note, the induced drag of canard-wing and wing-tail combinations is calculated for the limiting case in which the downstream surface is located in the Trefftz plane (infinite stagger), but based on the assumption that the loading on each surface is elliptical in isolation. It is shown that additional terms associated with induced circulation effects act to reduce the drag over that calculated using the classical theory. This induced thrust component is shown to be very small for the wing-tail configuration, but significant for the canard-wing layout. In addition, given elliptic loading on the upstream surface, an expression is derived for the loading distribution on the downstream surface which is optimum in the sense that the induced thrust component is maximized. The Trefftz plane assumption leads to a simplification of the analysis, since it means that the lift distribution of the downstream surface does not influence the drag of the upstream surface. The case of finite stagger, including the effects of bound vorticity on both surfaces, will be covered in a subsequent paper.

Comment on "Analytical Prediction of Vortex Lift"

S comments are warranted concerning the paper by J.W. Purvis entitled Analytical Prediction of Vortex Lift, which appears in the April 1981 issue of the Journal of Aircraft. The first comment is relative to the use of singular loadings in the design of mean camber shapes for wings experiencing leading-edge-vortex separated flow. By comparison, Ref. 1 outlines a low-drag, separated-flow design procedure which uses a more appropriate attached-flow initial loading that is nonsingular. There the final solution for the mean camber surface is obtained by iterating to find the shape-yielding minimum separated-flow drag at the required lift coefficient. The second comment is that the section-induced drag, see Eq. (12), does not have an elliptical distribution over the wing as proposed, but only in the Trefftz plane. This is well known (e.g., Ref. 2), and means that for most wings of interest the section-induced drag has to be determined in the near field. The consequence of this wrong assumption leads to an error in Eq. (26) and, hence, in the magnitude of the leading-edge suction force and vortex flow aerodynamics for wings with varying leading-edge sweep. The third comment deals with the missing multiplication by 2 in Eqs. (27), (30), (33), and (34), the missing division by 2 in Eqs. (28) and (37), and the typographical error in Eq. (32). Equation (32) should read

円形閉／開風胴中の翼の誘導抵抗

The spanwise lift distribution and induced drag of a wing in a circular wind tunnel, with either closed or open working section, are studied. The exact expressions of the spanwise optimum lift distribution and the minimum induced drag are obtained from the TREFFTZ plane Bow field in either case. In the closed tunnel case, as the ratio of wing span to tunnel diameter increases, the lift distribution changes gradually from elliptic to uniform one, and the induced drag also gradually decreases until it becomes zero. In the open tunnel case, the distortion of the lift distribution caused by the presence of the tunnel boundary is rather small, even when the ratio of wing span to tunnel diameter reaches unity. It is seen that the simple ROSENHEAD type of correction based on the elliptic lift distribution is satisfactory for all practical cases in which the lack of uniformity of the airstream or the size of the airfoil chord does not become excessive.

Higher order farfield drag minimization for a subcritical wing design code

A higher order Trefftz plane model of the undistorted interacting wing wakes of two symmetric subsonic planforms has been developed and integrated with an existing vortex lattice wing design code. The Trefftz plane calculation provides continuous, piecewise quadratically varying bound circulations, which are then interpolated in the nearfield vortex lattice representation of the wing. Integration of the surface slopes for a specified chord loading function provides the design wing camber surfaces. This paper outlines the theoretical development and presents comparisons between results obtained using the present theory and (1) previous exact solutions for minimum induced drag and (2) previous wing designs obtained using a discrete vortex wake moded.

Induced drag of a straight wing in a wind tunnel of circular cross section

The Induced drag of a wing with optimum lift distributions is studied, which lies on a diameter of a closed wind tunnel of circular cross section. Exact expressions of the spanwise lift distribution and the induced drag are obtained from the Trefftz plane flow field. When the ratio b/d of wing span to tunnel diameter increases, the lift distribution changes gradually from elliptic to uniform ones, and the induced drag also gradually decreases until it becomes zero. An alternative solution based on the method of matched asymptotic expansions confirms the exact theory to be valid. Throughout this paper the flow is assumed inviscid and incompressible, whereas the viscous effect are briefly descirbed.

Effect of flow shear on induced drag.

V Karman and Tsien formulated the lifting line theory for a wing in nonuniform flow in 1945. In their formulation, the approaching flow may have arbitrary variations in the plane perpendicular to the flight direction. By working with the perturbation pressure rather than the perturbation velocities, they have obtained the condition for the minimum induced drag and the method of calculation for the induced drag of a lifting line with arbitrary spanwise lift distribution. No specific examples, however, were given. In this note, we present the calculation of the induced drag of an elliptically loaded lifting line in an exponentially sheared flow in the vertical direction. The result applies equally well to a linearly sheared flow provided the shear rate is small. Consider a lifting line with a unit semispan in an incompressible, inviscid stream which is sheared in the vertical direction. Let the rectangular coordinate system be such that the y axis is in the direction of the span with the lifting line extending from y = — 1 to y = +1, the x axis in the direction of the approaching flow U(z), and the z axis pointing upwards. By considering the flow in the Trefftz plane (x -> oo ), von Karman and Tsien give the following governing equation: ] = 0 (1)

Theory of shear flow through a cascade

This paper considers aninviscid shear flow through a cascade of aerofoils spanning two parallel plane walls, on the assumptions of thin aerofoil theory. The spanwise distribution of lift coefficient on aerofoils in this type of flow, the flow features far downstream from the cascade, i.e. those in the Trefftz plane, and the induced drag due to the non-uniformity of the spanwise distribution of lift are studied theoretically, with some numerical examples.

A note on supersonic flow in the Trefftz plane

A note on supersonic flow in the Trefftz plane