Trefftz Plane Research Articles

Developing methods of decomposing the physical resistance components acting on a hydrofoil

The America’s Cup is a sailing competition contested by teams fielding bespoke yachts designed and built to the AC75 class regulations. These yachts make use of hydrofoils, which permit them to lift their hulls above the water surface. When in this “foiling” regime, the drag forces on the hull are drastically lower, permitting greater sailing speed. Predicting hydrodynamic forces produced from hydrofoils on race yachts remains challenging, due to a wide operating regime and complex hydrodynamical phenomena. In this study, existing wake survey methods of decomposing the total drag force of a hydrofoil into physical subcomponents were developed through examining their sensitivities and accuracy. Computational fluid dynamics (CFD) simulations of a rectangular NACA4412 hydrofoil were performed to generate data for the wake surveys. The hydrofoil was simulated at a submergence depth of one chord. Viscous flow was simulated with the k–ω SST turbulence model in steady state; and the volume-of-fluid (VOF) model was used to capture free surface effects. Chord Froude numbers of 0.5, 1, 2 and 4 were investigated, corresponding to Reynolds numbers of 2.57 × 105, 5.15 × 105, 1.03 × 106 and 2.06 × 106. Force decomposition was performed using two different wake survey methods – one each for the wave and viscous drag – as well as Trefftz plane analysis for the induced drag. Wave drag forces could be reliably isolated at all tested Froude numbers, while reliable predictions of the viscous drag were dependent on the Froude number and method of isolating the hydrofoil’s viscous wake region. Accurately isolating the induced drag component proved challenging due to free surface interactions. Careful selection of the wake survey location was also necessary for accurate predictions of any of the drag components.

Evaluating the effect of a ducted winglet on the induced drag of wind turbine blade using CFD and Trefftz plane analysis

A novel patented ducted winglet design is compared, using numerical simulations, to an unmodified wing tip and a standard winglet in the context of a wind turbine blade. The aim of the ducted winglet is to reduce the induced drag. This reduction of induced drag, at the wing tip, would be particularly beneficial to wind turbine performance. Throughout the comparison, we explore the aerodynamic phenomena described in the patent which claim to explain how the ducted winglet reduces induced drag. The simulations were run using a CFD solver, FLITE3D, that has been thoroughly validated on industrial scale aerodynamic problems. A mesh convergence study was carried out to ensure that the results have sufficient numerical accuracy for a consistent ranking with respect to the lift-to-drag ratio. A Trefftz plane analysis is conducted to measure the effect of the duct on induced drag. To the best of our knowledge, this is the first time that such a ducted design is tested in this context and compared to a standard unmodified blade and winglet. We found that the ducted winglet has a higher lift-to-drag ratio than an unmodified wing tip, but lower than a non-ducted winglet. Despite this, there is clear evidence that the duct had a positive influence on the induced drag. Our results show that with future optimisation this ducted winglet may prove beneficial to wind turbine design.

Estimating lift from wake velocity data in flapping flight

The application of the Kutta–Joukowski (KJ) theorem to estimating the lift of a flying animal based on wake velocity fields often leads to significant underprediction of the lift, which is known as the wake momentum paradox. This work attempts to answer the puzzling question on whether the KJ theorem is legitimate in its use for complex viscous unsteady wakes generated by flapping wings. The limitations in applying the KJ theorem to flapping wings are quantitatively examined through numerical simulations of viscous incompressible flows over three flapping wing models. The three flapping wing models studied in this work are a flapping wing with a fixed wingspan, a flapping wing with a dynamically changing wingspan and a dihedral flapping wing. The KJ theorem fails to give a satisfactory prediction of the time-averaged lift unless an effective span length is correctly computed. We propose a wake-sectional Kutta–Joukowski (WS-KJ) model to predict the time-averaged lift, where the effective span length is computed based on the time-averaged distance between the streamwise vorticity centroids in the right and left half sides of the Trefftz plane. The WS-KJ model incorporates the spatial evolutionary effects of the complex vortex structures in the wake and significantly improves the prediction of the time-averaged lift. The physical foundation for such improvement is explored. In addition, the time-dependent amplitude and phase changes of the unsteady lift are discussed as the fluid acceleration effect.

Fast multipole method applied to Lagrangian simulations of vortical flows

Lagrangian simulations of unsteady vortical flows are accelerated by the multi-level fast multipole method, FMM. The combination of the FMM algorithm with a discrete vortex method, DVM, is discussed for free domain and periodic problems with focus on implementation details to reduce numerical dissipation and avoid spurious solutions in unsteady inviscid flows. An assessment of the FMM-DVM accuracy is presented through a comparison with the direct calculation of the Biot–Savart law for the simulation of the temporal evolution of an aircraft wake in the Trefftz plane. The role of several parameters such as time step restriction, truncation of the FMM series expansion, number of particles in the wake discretization and machine precision is investigated and we show how to avoid spurious instabilities. The FMM-DVM is also applied to compute the evolution of a temporal shear layer with periodic boundary conditions. A novel approach is proposed to achieve accurate solutions in the periodic FMM. This approach avoids a spurious precession of the periodic shear layer and solutions are shown to converge to the direct Biot–Savart calculation using a cotangent function.

静止スラストを最大とするロータ設計法

In this paper, the practical design method of the maximum static thrust rotor based on the vortex method is described. After the vortex method is introduced, the simplified optimum rotor design method is presented. By analyzing the shape of the rotor wake, it was evident that the pitch of the vortices produced by the tip of the rotor is half that of the inboard vortices. The wake contraction rate parameter and axial vortex trajectory parameters are obtained from the tip vortices simulation. The accurate design method based on the vortex method with accurate wake model is then executed. The obtained maximum thrust rotor satisfies the conditions required on the Trefftz plane, which is not unfulfilled achieved by the simplified design method.

An efficient method for predicting zero-lift or boundary-layer drag including aeroelastic effects for the design environment

AbstractOne of the aerospace design engineer’s goals aims to reduce drag for increased aircraft performance, in terms of range, endurance, or speed in the various flight regimes. To accomplish this, the designer must have rapid and accurate techniques for computing drag. At subsonic Mach numbers drag is primarily a sum of lift-induced drag and zero-lift drag. While lift-induced drag is easily and efficiently determined by a far field method, using the Trefftz plane analysis, the same cannot be said of zero-lift drag. Zero-lift drag (CD,0) usually requires consideration of the Navier-Stokes equations, the solution of which is as yet unknown except by using approximate numerical techniques with computational fluid dynamics (CFD). The approximate calculation of zero-lift drag from CFD is normally computed with so-called near-field techniques, which can be inaccurate and too time consuming for consideration in the design environment. This paper presents a technique to calculate zero-lift and boundary-layer drag in the subsonic regime that includes aeroelastic effects and is suitable for the design environment. The technique loosely couples a two-dimensional aerofoil boundary-layer model with a 3D aeroelastic solver to compute zero-lift drag. We show results for a rectangular wing (baseline), a swept wing, and a tapered wing. Then compare with a rectangular wing with variable thickness and camber, thinning out from the root to tip (spanwise direction), thus demonstrating the practicality of the technique and its utility for rapid conceptual design.

Wing Tip Vortices from an Exergy-Based Perspective

The lens of exergy is used to investigate a wingtip vortex in the near wake over a range of angles of attack. Exergy is the measure of thermodynamically “available” energy as determined through the more discriminating second law of thermodynamics. Experiments were conducted in a water tunnel at Institute of Aerospace Systems at Aachen. The data were taken three chord lengths downstream in the Trefftz plane of an aspect ratio 5 Clark-Y wing with a square-edged wing tip using particle image velocimetry. Intuitively, the minimum available energy state is expected to correspond to the maximum lift-to-drag ratio angle of attack. This, however, is not the case here. Most interestingly, although only two-dimensional Trefftz plane data were used to obtain the exergy distribution across the individual wing-tip vortices, the crossover point for the out-of-plane change from wakelike to jetlike wing-tip vortex core axial flow (indicating the peak lift-to-drag ratio) is identified by the in-plane exergy distribution. This crossover point is not identifiable in the evaluation of any other characteristics calculated from in-plane quantities. Consequently, the exergy method holds promise as a metric for the improvement of aircraft performance through the reduction of lift-induced drag.

The aerodynamic cost of flight in the short-tailed fruit bat (Carollia perspicillata): comparing theory with measurement

Aerodynamic theory has long been used to predict the power required for animal flight, but widely used models contain many simplifications. It has been difficult to ascertain how closely biological reality matches model predictions, largely because of the technical challenges of accurately measuring the power expended when an animal flies. We designed a study to measure flight speed-dependent aerodynamic power directly from the kinetic energy contained in the wake of bats flying in a wind tunnel. We compared these measurements with two theoretical predictions that have been used for several decades in diverse fields of vertebrate biology and to metabolic measurements from a previous study using the same individuals. A high-accuracy displaced laser sheet stereo particle image velocimetry experimental design measured the wake velocities in the Trefftz plane behind four bats flying over a range of speeds (3-7 m s(-1)). We computed the aerodynamic power contained in the wake using a novel interpolation method and compared these results with the power predicted by Pennycuick's and Rayner's models. The measured aerodynamic power falls between the two theoretical predictions, demonstrating that the models effectively predict the appropriate range of flight power, but the models do not accurately predict minimum power or maximum range speeds. Mechanical efficiency--the ratio of aerodynamic power output to metabolic power input--varied from 5.9% to 9.8% for the same individuals, changing with flight speed.

Drag Characteristics for Optimally Span-Loaded Planar, Wingletted, and C Wings

This paper focuses on the drag characteristics of optimally span-loaded planar, wingletted, and C wings. The span load is optimized resulting in minimum induced or total drag. The wing-root bending moment is kept constant for all analyzed wings to ascertain that different wings have comparable weight. The optimum span loadings for the different types of wing are calculated using a fast and simple numerical method. The wings are analyzed in the Trefftz plane, infinitely far behind the wing. Lagrange multipliers are used to calculate the optimum span loading resulting in minimum induced or total drag, with the wing-root bending moment and/or the lift coefficient as constraint. The induced drag can be calculated using the optimum span loading. The profile drag is assumed to be a function of the local lift coefficient. The results indicate that the C wing does not have real aerodynamic performance advantages compared to a wingletted wing. For wings with span and/or aspect ratio constraints, a winglet offers a drag reduction relative to a planar wing.

Time-resolved wake structure and kinematics of bat flight

We present synchronized time-resolved measurements of the wing kinematics and wake velocities for a medium sized bat, Cynopterus brachyotis, flying at low-medium speed in a closed-return wind tunnel. Measurements of the motion of the body and wing joints, as well as the resultant wake velocities in the Trefftz plane are recorded at 200 Hz (approximately 28–31 measurements per wing beat). Circulation profiles are found to be quite repeatable although variations in the flight profile are visible in the wake vortex structures. The circulation has almost constant strength over the middle half of the wing beat (defined according the vertical motion of the wrist, beginning with the downstroke). A strong streamwise vortex is observed to be shed from the wingtip, growing in strength during the downstroke, and persisting during much of the upstroke. At relatively low flight speeds (4.3 m/s), a closed vortex structure behind the bat is postulated.

Direct measurements of the kinematics and dynamics of bat flight

Experimental measurements and analysis of the flight of bats are presented, including kinematic analysis of high-speed stereo videography of straight and turning flight, and measurements of the wake velocity field behind the bat. The kinematic data reveal that, at relatively slow flight speeds, wing motion is quite complex, including a sharp retraction of the wing during the upstroke and a broad sweep of the partially extended wing during the downstroke. The data also indicate that the flight speed and elevation are not constant, but oscillate in synchrony with both the horizontal and vertical movements of the wing. PIV measurements in the transverse (Trefftz) plane of the wake indicate a complex ‘wake vortex’ structure dominated by a strong wing tip vortex shed from the wing tip during the downstroke and either the wing tip or a more proximal joint during the upstroke. Data synthesis of several discrete realizations suggests a ‘cartoon’ of the wake structure during the entire wing beat cycle. Considerable work remains to be done to confirm and amplify these results.

Quadrilateral and triangular plane elements with rotational degrees of freedom based on the hybrid Trefftz method

In this paper, we suggest flexible, accurate, and robust linear elastic plane elements. We introduce drilling degrees of freedom to the hybrid Trefftz plane element by using Allman's quadratic displacement instead of conventional finite element approximation for the boundary displacement. Added drilling degrees increase the accuracy and robustness to mesh quality though they keep the advantages of the hybrid Trefftz method such as efficiency and freedom of the shape of the element. In addition, due to the hybrid Trefftz formulation, Poisson's locking, which may come from Allman's quadratic displacement, disappears.

A hybrid Trefftz plane elasticity element with drilling degrees of freedom

We suggest a linear plane element that is accurate and easy to use. The new element is based on the hybrid Trefftz method. To give the drilling degrees of freedom and to enhance the performance of the element, we take the hybrid Trefftz method based on the total potential energy principle and adopt Allman’s quadratic displacement field as a boundary displacement field. To stabilize the drilling degrees of freedom, we restrict the rotation field by a penalty term. The penalty term ties up the drilling rotation with the true rotation derived from the internal displacement field, which is made up of Trefftz functions. With the proposed formulation, we can make up polygonal elements in a unified framework. The performance of the new element reveals an advantage in modeling and analyzing, as well as good accuracy and robustness.

Effect of counter-rotating vortices on the development of aircraft wakes

AbstractA two-dimensional vortex particle model is used for studying the development of the vortex wake generated by the wing of an aircraft in the Trefftz plane. Two-dimensional, finite-area vortex structures simulate the initial vorticity distribution at a near-field cross-section of the wing, as provided by either measurements or simulations. The code is used for studying the effect of weak or strong counter-rotating vortices on the development of an aircraft wake. Application in a three-vortex configuration, consisting of the tip and flap vortices, plus a weak negative vorticity sheet lying between them, supports the hypothesis of Graham that the merging of the tip and flap vortices is prevented, because in such an arrangement the counter-rotating vorticity sheet is wrapped around the primary vortices. The present results indicate that even in the far field the tip and flap vortices remain distinct and follow a helical trajectory with large-pitch and small-radius. The code is next applied to the two-vortex system of Savas’s triangular wing, in which the circulation of the flap vortex is comparable to that of the tip. Although a 2D analysis is inappropriate for stability analysis it is still useful for a quick qualitative investigation. Results indicate that the flap and tip vortices follow a helical trajectory with large-pitch but also very large radius. During one period, the flap vortex covers a span wise distance equal to the wing span. Such a flow has not been observed in flight or in laboratory tests. Actually, Ortega et al, who studied experimentally the triangular wing of O. Savas, found out that before concluding one full spiral, the vortices are literally destroyed (rapid spreading of their vorticity) by an instability mechanism.

Aerodynamic Interference of a Large and a Small Aircraft.

The influence of wake from a leading aircraft, Boeing 757, on a following aircraft, Cessna 750 Citation X, is investigated using the unsteady vortex lattice method. The effect of the wakes from both wings and each other is considered and the new position of the wakes from both aircraft calculated. As the midspan of the Cessna wing approaches the Boeing 757 trailing vortex core, the roll moment increases to its maximum value. Strong changes appear in aerodynamic loads of the Cessna as it moves near the Boeing 757’s wake. Rapid change occurs as the Cessna moves in transverse or vertical directions near the Boeing’s wake. The unsteady aerodynamic loads on wings moving along different paths are calculated. The Boeing 757’s wake strongly affects the aerodynamic loads and the wake of the Cessna. Also included in the present results are numerical simulations of wakes and velocity vectors in the Trefftz plane detailing the influence of the wake from the leading aircraft, the Boeing 757.

Optimization of Wind Turbines Using Helicoidal Vortex Model

The problem of the design of a wind turbine for maximum output is addressed from an aerodynamical point of view. It is shown that the optimum inviscid design, based on the Goldstein model, satisfies the minimum energy condition of Betz only in the limit of light loading. The more general equation governing the optimum is derived and an integral relation is obtained, stating that the optimum solution satisfies the minimum energy condition of Betz in the Trefftz plane “in the average.” The discretization of the problem is detailed, including the viscous correction based on the 2-D viscous profile data. A constraint is added to account for the thrust on the tower. The minimization problem is solved very efficiently by relaxation. Several optimized solutions are calculated and compared with the National Renewable Energy Laboratory (NREL) rotor, using the same profile, but different chord and twist distributions. In all cases, the optimization produces a more efficient design.

Theoretical Solutions for Finite-Span Wings of Arbitrary Shapes Using Velocity Singularities

This paper presents a new method of solution for the e nite-span wings of arbitrary shapes, which avoids the dife culties of the previous methods. This method uses velocity singularities in the Trefftz plane to derive the contributions in the solution of the circulation distribution caused by the changes in the spanwise variation of the wing chord and incidence. The new functions derived for these contributions contain both natural and forced symmetry and antisymmetry terms (which are absent in the previous methods ) and thus represent a correct mathematical modeling of the physical problemsthat lead to highly accurate theoretical solutions. The method has beenvalidatedbycomparisonwiththesolutionsobtainedbyRasmussenandSmithandCarafoliforrectangularand tapered wings of uniform incidence, and with the panel method results by Katzand Plotkin. Accurateand efe cient theoretical solutions of aeronautical interest are then obtained for wings with asymmetric incidence distributions caused by symmetric and antisymmetric dee ections of e aps and ailerons and for wings with curved leading and trailing edges and variable incidence, which are dife cult to model in current panel methods.

Spanwise Optimum Normal Force Distribution of a Hemi-Circular Wing-in-Ground.

The aerodynamic characteristics of a hemi-circular wing-in-ground are studied analytically. The exact expression of a spanwise optimum normal force distribution and the minimum induced drag are obtained from Trefftz plane flow field analysis, where Sohngen’s inversion integral formula and the elliptic integrals and Jacobian elliptic functions are used. The minimum induced drag obtained in the present study is coincident with that of Mamada and Ando.

Mathematical modeling of the columnar vortex

The columnar laminar vortex has a two-zone structure in the case of a highly swirling flow. Various asymptotics for the laminar, wave, and turbulent flows in the Trefftz plane are discussed.

Lift Prediction of Spanwise Cambered Delta Wings

An analytic prediction method is derived to estimate the lift of delta wings with constant anhedral or dihedral, i.e., a V wing. The method is based on the leading-edge suction analogy, with expressions for leading-edge suction being developed that account for anhedral and dihedral. An empirical correction is combined with the resulting expression to incorporate an effective leading-edge sweep interpretation of wing roll. This approach allows an estimation of the disparate effects of anhedral and dihedral. The attached e ow lift component is estimated using a Trefftz plane expression for the apparent mass of a V wing. Comparisons are made with a variety of planforms, with encouraging agreement between theory and experiment being demonstrated.