Lifting Line Theory Research Articles

Leading-Edge Vortex as a High-Lift Mechanism for Large-Aspect-Ratio Wings

Enabling a leading-edge vortex (LEV) is a possible mechanism to significantly increase the lift on wings. This is well known for slender delta wings, for which various analytical models were developed and show accurate predictions of the lift enhancement. This paper shows that, when suitably designing the wing planform, LEVs can also exist on high-aspect-ratio (high-AR) wings. In this study, a quasi-three-dimensional flow model is presented for solving the stationary LEV phenomenon over a high-AR swept back wing, with sweep increasing toward the wingtip. Our model captures the three-dimensional phenomenon by satisfying conservation of mass and vorticity within the LEV, and using a combination of strip theory and the lifting-line theory. Our model predictions are compared with flow visualization data on a parabolic swept back wing. Results confirm that suitable sweep wing geometry can fix an LEV steadily over the upper wing surface, resulting in significant lift enhancement of up to 70%.

Adverse-Yaw Control During Roll for a Class of Optimal Lift Distributions

Lifting-line theory has formed the basis of wing-design understanding for nearly a century. This theory has previously been used to develop a class of optimal lift distributions that minimize induced drag for finite wings under specific structural and aerodynamic constraints. This class of lift distributions was first introduced by Prandtl in 1933 and includes the elliptic lift distribution and the bell-shaped lift distribution, among others, as special cases. Here, it is shown that, for all lift distributions in this class except the elliptic lift distribution, an antisymmetric twist distribution can be added to the wing to control the adverse yawing moment during roll. The requisite amount of this additional twist distribution depends on the prescribed yawing-moment sign and magnitude, and it can be used to produce zero adverse yaw or even proverse yaw during roll. Of course, employing this methodology to control the roll–yaw coupling produces additional induced drag relative to the case without roll–yaw coupling control. This induced-drag penalty is studied in detail and shows that a symmetric lift distribution within this class can be selected to minimize induced drag for a desired roll–yaw coupling. Roll initiation and steady rolling rate are both considered. Results can be applied to the design of morphing aircraft to minimize the use of a vertical control surface for yaw control.

Takeoff Simulation of Air Taxi by Using Navier–Stokes Equations

Takeoff trajectory computations are made for a typical small transport aircraft suitable for an urban air taxi. The selected wing–body model consists of lifting and pushing propellers. Flow is modeled using the Navier–Stokes equations. A procedure is developed to embed the trajectory motion equations in an overset grid topology along with rotating blades. Results are validated with lifting line theory computations. A typical takeoff scenario is demonstrated by time-accurately integrating trajectory equations with the flow equations. The present work extends the capabilities of the current Navier–Stokes solvers to simulate trajectory motions of urban air taxi configurations.

The design of a small lab-scale wind turbine model with high performance similarity to its utility-scale prototype

This paper discussed the procedure of an optimization design method involving a small lab-scale wind turbine rotor. To simulate the aerodynamic performance of a prototype wind turbine at utility-scale by scaled model experiments (especially, the visualization measurements of wake flow in small wind tunnels), an appropriate similar design of the small lab-scale rotor is needed. However, the similar design of blade performance from a prototype wind turbine for small scaled model is rare. In this paper, using a 2.5 MW utility-scale wind turbine as the prototype, we set up a new optimization process of similar design for a model wind turbine with a 320 mm rotor based on the Lifting-Line Theory (LLT) with wake-induced corrections. Results show a dramatic deviation in performance of the geometric scaled model compared to the objective values, which demonstrates the significance of the Reynolds number effects. With the optimized blade, the distribution of normal thrust force is very similar to the objective values. Geometric characteristics variations along the blade span are similar to the prototype. With the tip speed ratio runs at a roughly matched rated condition, the optimized rotor gives a better performance than other models used in previous wind tunnel studies.

Numerical Study on Flow Around Modern Ship Hulls with Rudder-Propeller Interactions

Reducing the fuel consumption of ships presents both economic and environmental gains. Although in the past decades, extensive studies were carried out on the flow around ship hull, it is still difficult to calculate the flow around the hull while considering propeller interaction. In this paper, the viscous flow around modern ship hulls is computed considering propeller action. In this analysis, the numerical investigation of flow around the ship is combined with propeller theory to simulate the hull-propeller interaction. Various longitudinal positions of the rudder are also analyzed to determine the effect of rudder position on propeller efficiency. First, a numerical study was performed around a bare hull using Shipflow computational fluid dynamics (CFD) code to determine free-surface wave elevation and resistance components. A zonal approach was applied to successively incorporate “potential flow solver” in the region outside the boundary layer and wake, “boundary layer solver” in the thin boundary layer region near the ship hull, and “Navier-Stokes solver” in the wake region. Propeller open water characteristics were determined using an open-source MATLAB code OpenProp, which is based on the lifting line theory, for the moderately loaded propeller. The obtained open water test results were specified in the flow module of Shipflow for self-propulsion tests. The velocity field behind the ship was recalculated into an effective wake and given to the propeller code that calculates the propeller load. Once the load was known, it was transferred to the Reynolds-averaged Navier-Stokes (RANS) solver to simulate the propeller action. The interaction between the hull and propeller with different rudder positions was then predicted to improve the propulsive efficiency.

Analytic and computational analysis of wing twist to minimize induced drag during roll

Geometric and/or aerodynamic wing twist can be used to produce a lift distribution that results in a rolling moment. A decomposed Fourier-series solution to Prandtl’s lifting-line theory is used to develop analytic spanwise antisymmetric twist distributions for roll control that minimize induced drag on wings of arbitrary planform in pure rolling motion. Roll initiation, steady rolling rate, and the transition between the two are each considered. It is shown that if these antisymmetric twist distributions are used, the induced drag is proportional to the square of the rolling moment, and the induced drag during a steady rolling rate is equal to that on the wing at the same lift coefficient with no rolling rate or antisymmetric twist distribution. Results also show that if these antisymmetric twist distributions are used on straight, tapered wings without symmetric twist, any rolling maneuver for which the rolling rate and rolling moment have the same sign will always produce a yawing moment in the opposite direction. Computational results are also included, which were obtained using a gradient-based optimization algorithm in combination with a modern numerical lifting-line algorithm to find the optimum twist solutions. The resulting twist, induced drag, and yawing moment solutions compare favorably with the analytic solutions developed in the text. The solutions presented here can be used to inform the design of morphing aircraft.

Aerodynamic Optimization of a UAV Wing subject to Weight, Geometric, Root Bending Moment, and Performance Constraints

In this study, the optimization of a low-speed wing with functional constraints is discussed. The aerodynamic analysis tool developed by the coupling of the numerical nonlinear lifting-line method to Xfoil is used to obtain lift and drag coefficients of the baseline wing. The outcomes are compared with the results of the solver based on the nonlinear lifting-line theory implemented into XLFR5 and the transition shear stress transport model implemented into ANSYS-Fluent. The agreement between the results at the low and moderate angle of attack values is observed. The sequential quadratic programming algorithm of the MATLAB optimization toolbox is used for the solution of the constrained optimization problems. Three different optimization problems are solved. In the first problem, the maximization of CL3/2/CD is the objective function, while level flight condition at maximum CL3/2/CD is defined as a constraint. The functional constraints related to the wing weight, the wing planform area, and the root bending moment are added to the first optimization problem, and the second optimization problem is constructed. The third optimization problem is obtained by adding the level flight condition and the available power constraints at the maximum speed and the level flight condition at the minimum speed of the baseline unmanned air vehicle to the second problem. It is demonstrated that defining the root bending moment, the wing area, and the available power constraints in the aerodynamic optimization problems leads to more realistic wing planform and airfoil shapes.

Experimental Investigation of High-Angle-of-Attack Aerodynamics of Low-Aspect-Ratio Rectangular Wings Configured with NACA0012 Airfoil Section

To examine the high-angle-of-attack (AOA) aerodynamics, the conventional lift and drag were measured in one revolution AOA by the dynamic load cell in the wind tunnel for rectangular wings of the NACA0012 section with four different aspect ratios, 3, 4, 5, and 6, at a Reynolds number of 1.0 × 105. The results were analyzed in the normal and reverse modes of the airfoil. It was found that the reverse airfoil is disadvantageous to the lifting device because of the earlier stall than the normal and the substantial drag increases before the stall. In the entire AOA range, Prandtl’s lifting line theory seems to be applicable in general, but the profiles of the lift coefficient are not linear anymore. It was also found that the drag coefficient of the normal airfoil mode is affected by the delta wing-type vortex wrap as well as the downwash, and that the downwash effect was dominated between the deep stall and the second peak. Using the expanding scales, which have an exponential decay rate with the aspect ratio, the polar plots of the four different wings overlap in one circle with a radius of 1.0 at the same origin.

Aerodynamic model of propeller–wing interaction for distributed propeller aircraft concept

The present investigation addresses two key issues in aerodynamic performance of a propeller–wing configuration, namely linear and nonlinear predictions with low-order numerical models. The developed aerodynamic model is targeted to be used in the preliminary aircraft design loop. First, the combination of selected propeller model, i.e. blade element theory with the wing model, i.e. lifting line theory and vortex lattice method is considered for linear aerodynamic model. Second, for the nonlinear prediction, a modified vortex lattice method is paired with the two-dimensional viscous effect of the airfoils to simplify and reduce the computational time. These models are implemented and validated with existing experimental data to predict the differences in lift and drag distribution. Overall, the predicted results show agreement with low percentage of error compared with the experimental data for various thrust coefficients and produced induced drag distribution that behaves as expected.

Design optimization of a curved wind turbine blade using neural networks and an aero-elastic vortex method under turbulent inflow

This article describes the application of neural networks for the design optimization of a curved wind turbine blade using an aero-elastic simulator with synthetic inflow turbulence. A vortex particle method where the wind turbine blades are represented by lifting-line theory is used, while the wind turbine structural dynamics are modeled using a finite-element multi-body based approach. A neural network together with a gradient-based optimizer allows to quickly design a new curved wind turbine blade in a complex aero-elastic wind-turbine simulation scenario. The blade design found from the neural network has increased pre-bend and sweep compared to the straight blade design. It produces approximately 1% more power on average with a slight increase of mean thrust on the rotor of 0.02% compared to the straight one. This study demonstrates that neural networks can be effective for designing wind turbine rotor blades involving complex aero-elastic simulation scenarios with turbulent inflow conditions. Further work may improve the performance of the neural network's predictive capabilities as well as the optimized design.

Revisiting the combinatorics of lifting line and 2D vortex lattice theory

ABSTRACTThis work revisits analyses of lifting line theory by A.R. Collar from 1958, bringing to bear some modern tools and techniques. The 2D vortex lattice model is also considered. Interesting combinatorial properties and simplified expressions for terms are presented, a simplified proof of model convergence shown, extension of the convergence properties to elliptical and arbitrary wing planforms is demonstrated, and approximations provided. An alternative proof of the optimality of constant downwash is presented. Modern automated theorem proving techniques are employed in confirming the combinatorial results.

Hydrodynamic Shape Optimization of an Autonomous Underwater Vehicle With an Integrated Lifting Line and Viscous Continuous Adjoint Approach

The designs of the optimal shape of an autonomous underwater vehicle (AUV) and its aft propeller's optimal circulation can interact with each other. We propose an optimization approach that integrates the lifting line theory and the continuous adjoint method to address this conflict. The optimal circulation of a propeller is obtained via lifting line theory. The corresponding propeller force is introduced into the Reynolds-averaged flow equations as a source term and is considered in the development of a continuous adjoint formulation for the AUV shape optimization. Adjoint equations, with boundary conditions and sensitivity derivative expressions for the drag reduction problem, are derived in detail. The shapes of a small AUV and its aft propeller are simultaneously optimized using the proposed approach. Numerical results show that the proposed method decrease AUV drag by 17% at a rated speed. And a prototype test further confirms the effectiveness of the method.

A strongly-coupled model for flexible rotors

A fluid–structure model describing the equilibrium state of a flexible blade rotor with its own wake is derived for various external axial flow conditions. The model is based on three building blocks. The two-dimensional lifting-line theory is first used to compute the local aerodynamic loads and the blade circulation profile. The blade deformation is then obtained by solving the nonlinear equations for bending and twisting angles deduced from a one-dimensional beam model. Finally, the wake is obtained using a Joukowski model. In this wake model, the wake of each blade is modeled by two small-core-size counter-rotating vortices emitted from the rotor axis and blade tip. The velocity field induced by these vortices is computed using the Biot–Savart law. We show that, in the rotor frame, we can obtain a stationary vortex structure for almost any vertical flight regimes. This wake solution can then be used to compute the induced velocity in the rotor plane and apply the two-dimensional lifting-line theory again. By iterating a few times this loop, we converge toward a nonlinear solution of the problem for which the aerodynamics loads, blade deformation and wake structure are compatible.As illustration, this newly-developed model is applied to two rotors. We analyze the effects of the external wind conditions, geometry and material properties of the blades on the blade deformation and wake characteristics. We show that we can describe slow descending regimes for which the classical momentum theory does not apply.

Experimental and theoretical investigations on the effect of a single leading-edge protuberance on airfoil performance

Geometrical modification of airfoils by leading-edge protuberances has attracted plenty of attention as a passive separation control method. In this research, the effect of a single leading-edge protuberance with different amplitudes on the performance of a two-dimensional airfoil (NACA 634-021) is investigated through wind tunnel experiment and theoretical analysis. Force measurement experiment shows that all the modified airfoils demonstrate a special two-step stall performance. When the first step of stall occurs, the lift coefficient of the modified airfoils drops to a medium value between the pre-stall and post-stall lift coefficients of the baseline airfoil and keeps nearly constant within a range of AOA (angle of attack). Tuft visualization reveals that during the first step of stall, the flow pattern on different sides of the protuberance is distinctly different, respectively, showing leading-edge separation and trailing-edge separation, which are compartmentalized by the attached flow originated from the peak of the protuberance. The flow control mechanism is contributed to the combined effect of the early occurrence of the leading-edge separation around the protuberance root, the inherent hysteretic performance, and the flow compartmentalization effect. A theoretical model based on the lifting line theory, which takes the amendment of the hysteresis and compartmentalization effects into consideration, is proposed to analyze the performance of the modified airfoils. The sectional and overall performances by theoretical analysis are highly consistent with the tendency of the experiment results. Based on the theoretical model, the spanwise circulation gradient induced by the protuberance is considered to be responsible for the earlier separation around the protuberance roots at pre-stall AOAs. Further investigation on the effect of the geometrical parameters of the protuberance reveals that the amplitude-to-wavelength ratio plays a dominant role on the performance of the modified airfoils. The special flow phenomenon induced by the single leading-edge protuberance discovered in this research and the proposed mechanism provide an essential point of view for fully understanding the combined mechanism of multiple protuberances.

Roll Control of Low-Aspect-Ratio Wings Using Articulated Winglet Control Surfaces

The use of tip mounted winglets with independently variable cant angles was investigated as a means of roll control on wings with an aspect ratio of one. Wind-tunnel testing was performed in which a six-axis force balance was used to measure the total aerodynamic load on wings with winglet control surfaces. Stereoscopic digital particle image velocimetry of the near-wake plane was used to show how the topology of the tip vortices changed with winglet deflection. Shifts in the location of the right tip vortex core are considered to be responsible for roll moment generation because they indicate changes in the symmetry of suction-side flow structures. All winglet deflections were observed to shift the right tip vortex core inboard, and thereby shorten the effective span of the wing. The effect of a winglet deflection may be approximated as a change in the wing aspect ratio and a lateral shift in the wing aerodynamic center. Prandtl’s lifting line theory provides a closed-form estimate for the reduction in lift caused by a winglet deflection. A geometrical argument was made to account for the induced roll moment. The right tip vortex core also shifts vertically, following the deflected wing tip. Vertical shifts in the right tip vortex result in an angle between the wing span line and a line connecting the two tip vortices. A positive angle is defined as the right tip vortex higher over the wing than the left, and it is accompanied by a positive roll moment. While in sideslip, the wing with no winglet deflection experiences a considerable roll moment as a result of a vertical and lateral shift in the two tip vortices. The articulated winglets are observed to partially mitigate these effects when the upstream winglet is actuated, and thus show promise as a direct means of disturbance rejection.

Filtered lifting line theory and application to the actuator line model

Lifting line theory describes the cumulative effect of shed vorticity from finite span lifting surfaces. In this work, the theory is reformulated to improve the accuracy of the actuator line model (ALM). This model is a computational tool used to represent lifting surfaces, such as wind-turbine blades in computational fluid dynamics. In ALM, blade segments are represented by means of a Gaussian body force distribution with a prescribed kernel size. Prior analysis has shown that a representation of the blade using an optimal kernel width$\unicode[STIX]{x1D716}^{opt}$of approximately one quarter of the chord size results in accurate predictions of the velocity field and loads along the blades. Also, simulations have shown that use of the optimal kernel size yields accurate representation of the tip-vortex size and the associated downwash resulting in accurate predictions of the tip losses. In this work, we address the issue of how to represent the effects of finite span wings and tip vortices when using Gaussian body forces with a kernel size larger than the optimal value. This question is relevant in the context of coarse-scale large-eddy simulations that cannot afford the fine resolutions required to resolve the optimal kernel size. For this purpose, we present a filtered lifting line theory for a Gaussian force distribution. Based on the streamwise component of the vorticity transport equation, we develop an analytical model for the induced velocity resulting from the spanwise changes in lift force for an arbitrary kernel scale. The results are used to derive a subfilter-scale velocity model that is used to correct the velocity along the blade when using kernel sizes larger than$\unicode[STIX]{x1D716}^{opt}$. Tests are performed in large-eddy simulation of flow over fixed wings with constant and elliptic chord distributions using various kernel sizes. Results show that by using the proposed subfilter velocity model, kernel-size independent predictions of lift coefficient and total lift forces agree with those obtained with the optimal kernel size.

Modeling and control of parafoils based on computational fluid dynamics

• Leading-edge incision and trailing-edge deflection are incorporated into the calculation of aerodynamic coefficients . • A flexible dynamic model with eight degrees of freedom is established. • A characteristic modeling method is used to reduce the order of the dynamic model. • Generalized predictive control based on a characteristic model is applied to the parafoil system. The determination of aerodynamic parameters of parafoil canopies has been a crucial issue because it affects the model precision. To calculate the aerodynamic coefficients of a canopy, the lifting-line theory has been used in the traditional method. However, because of the existence of leading-edge incisions, there are some restrictive assumptions in lifting-line theory when one is calculating the aerodynamic coefficients of a canopy. Therefore in this article we calculate the aerodynamic coefficients on the basis of computational fluid dynamics . As an improvement, the effects of a leading-edge incision and trailing-edge deflection are considered. Firstly, lift and drag coefficients are obtained by use of computational fluid dynamics. Then the least-squares method is used to identify incision and deflection factors. Furthermore, an eight-degrees-of-freedom mathematical model of a parafoil system is established on the basis of the parameters obtained. Finally, a novel control algorithm, generalized predictive control based on a characteristic model, is applied to the system. The precision of the model established and the effectiveness of the proposed control method are validated by simulation and airdrop testing.

Source term model for rod vortex generator

PurposeThe simulations of grid-resolved rod vortex generators (RVGs) require high computational cost and time. Additionally, the computational mesh topology must be adjusted to rods geometries. The purpose of this study is to propose the new source term model for RVG.Design/methodology/approachThe model was proposed by modification of Bender, Anderson, Yagle (BAY) model used to predict flows around different type of vortex generators (VGs) – vanes. Original BAY model was built on lifting line theory. The proposed model was implemented in ANSYS Fluent by means of the user-defined function technique. Additional momentum and energy sources are imposed to transport equations.FindingsThe computational results of source term model were validated against experimental data and numerical simulation results for grid-resolved rod. It was shown that modified BAY model can be successfully used for RVG in complex cases. An example of BAY model application for RVG on transonic V2C airfoil with strongly oscillating shock waves is presented. Aerodynamic performance predicted numerically by means of both approaches (grid resolved RVG and modeled) is in good agreement, what indicates application opportunity of the proposed model to complex cases.Practical implicationsModified BAY model can be used to simulate the influence of RVGs in complex real cases. It allows for time/cost reduction if the location or distribution of RVG has to be optimized on a profile, wing or in the channel.Originality/valueIn the paper, the new modification of BAY model was proposed to simulate RVGs. The presented results are innovative because of original approach to model RVGs.

Combined Blade-Element Momentum—Lifting Line Model for Variable Loads on Downwind Turbine Towers

Downwind rotors are a promising concept for multi-megawatt scale large wind turbines due to their advantages in safety and cost reduction. However, they have risks from impulsive loads when one of the blades passes across the tower wake, where the wind speed is lower and locally turbulent. Although the tower shadow effects on the tower loads have been discussed in former studies, there is currently no appropriate model for the blade-element and momentum theory so far. This study formulates the tower shadow effects on the tower load variation induced by blades using the lifting line theory, which does not require any empirical parameters. The method is verified via computational fluid dynamics for a 2 MW(megawatt), 3-bladed downwind turbine. The amplitude and the phase of the variation are shown to be accurate in outboard sections, where the rotor-tower clearance is large (>3.0 times of the tower diameter) and the ratio of the blade chord length is small (<0.5 times of the tower diameter), in both of rated and cut-out conditions.

Systematic design methodology for development and flight testing of a variable pitch quadrotor biplane VTOL UAV for payload delivery

This paper discusses the conceptual design and proof-of-concept flight demonstration of a novel variable pitch quadrotor biplane Unmanned Aerial Vehicle concept for payload delivery. The proposed design combines vertical take-off and landing (VTOL), precise hover capabilities of a quadrotor helicopter and high range, endurance and high forward cruise speed characteristics of a fixed wing aircraft. The proposed methodology is described by taking an example of a VTOL Unmanned Aerial Vehicle (UAV) for a mission requirement of carrying and delivering 6 kg payload to a destination at 16 km from the point of origin. First, the design of proprotors is carried out using a physics based modified Blade Element Momentum Theory (BEMT) analysis, which is validated using experimental data generated for the purpose. Proprotors have conflicting requirement for optimal hover and forward flight performance and a compromise solution has to be chosen. Next, the biplane wings are designed using simple lifting line theory. The airframe design is followed by power plant selection and transmission design. Finally, weight estimation is carried out to complete the design process. The proprotor design with 24° preset angle and −24∘ twist is found to be suitable for the current mission profile, based on 70% weightage to forward flight and 30% weightage to hovering flight conditions. A reduction in operating RPM of the proprotors from 3200 during hover to 2000 during forward flight results in 27% increase in proprotor efficiency. The VTOL UAV design proposed in this paper is also predicted to consume 64% less power in airplane mode compared to the quadrotor mode, thereby establishing the benefit of this hybrid concept. A proof-of-concept scaled prototype is fabricated using commercial-off-the-shelf parts and a Proportional Integral Derivative (PID) controller is developed and implemented on open source PixHawk autopilot board to enable stable hovering flight and attitude tracking during hover flight tests.