Lifting Line Theory Research Articles

A Solution Method for the Filtered Lifting Line Theory

A Solution Method for the Filtered Lifting Line Theory

Benchmarking of Aerodynamic Models for Isolated Propellers Operating at Positive and Negative Thrust

Operating a conventional propeller at negative thrust results in the operation of positively cambered blade sections at negative angles of attack, leading to flow separation. Consequently, accurately simulating the aerodynamics of propellers operating at negative thrust poses a greater challenge than at positive thrust. This study offers a comprehensive assessment of the aerodynamic modeling capabilities of numerical methods, spanning low to high fidelity, for computing propeller performance across both positive and negative thrust regimes. Low-fidelity methods, namely, blade-element momentum and lifting line theories, effectively predict propeller performance trends at positive thrust. However, they fail to capture trends at negative thrust beyond the maximum power output point due to the neglect of three-dimensional flow effects. Both steady and unsteady Reynolds-averaged Navier–Stokes (RANS) simulations with y+<1 perform well across both positive and negative thrust conditions, with errors below 2% for both thrust and power magnitudes near the maximum power output point. Lattice-Boltzmann very-large-eddy simulations (LB-VLESs) with y+≤10 exhibit excellent agreement with experimental data with less than 1% error near the maximum power output point but with significant computational costs. Conversely, LB-VLESs with y+≥15 offer a more economical approach to capture general trends with the computational cost of the same order as unsteady RANS. However, wall models introduce errors in modeling flow separation, leading to a 16% overestimation of power magnitude near the maximum power output point. The results highlight the necessity of using tools with increased fidelity levels when considering propeller operation at negative thrust compared to the conventional positive thrust regime.

On the power and control of a misaligned rotor – beyond the cosine law

Abstract. We present a new model to estimate the performance of a wind turbine operating in misaligned conditions. The model is based on the classic momentum and lifting-line theories, considering a misaligned rotor as a lifting wing of finite span, and accounts for the combined effects of both yaw and uptilt angles. Improving on the classical empirical cosine law in widespread use, the new model reveals the dependency of power not only on the misalignment angle, but also on some rotor design parameters and – crucially – on the way a rotor is governed when it is yawed out of the wind. We show how the model can be readily integrated with arbitrary control laws below, above, and around the rated wind speed. Additionally, the model also shows that a sheared inflow is responsible for the observed lack of symmetry for positive and negative misalignment angles. Notwithstanding its simplicity and insignificant computational cost, the new proposed approach is in excellent agreement with large eddy simulations (LESs) and wind tunnel experiments. Building on the new model, we derive the optimal control strategy for maximizing power on a misaligned rotor. Additionally, we maximize the total power of a cluster of two turbines by wake steering, improving on the solution based on the cosine law.

Dynamic hydroelasticity of composite appendages with reverse-mode algorithmic differentiation

Composite materials enable the tailoring of load-dependent, passive, shape adaptation because of the directional stiffness of the fibers. However, excessive flow-induced vibrations and dynamic instabilities represent a design challenge for composite marine appendages. We develop DCFoil, a dynamic composite foil solver that uses one-dimensional models—composite beam elements and unsteady lifting line theory—to perform static and dynamic frequency domain analysis. The program is differentiated to provide derivatives of an aggregated flutter function with respect to design variables. The flutter function characterizes dynamic hydroelastic stability. We apply the solver to investigate the hydroelastic performance of a composite fin bulb keel based on the IMOCA 60 class of racing yachts, which were reported to have flutter problems. Based on the derivatives computed for this bulb keel, we found that the foil thickness has the most significant impact on flutter speed. This work shows both the development of and utility of a program with flutter derivatives for flexible composite marine appendage design.

Aero-propulsion analysis of distributed ducted-fan propulsion based on lifting-line driven body-force model

As the environmental problems become increasingly serious, distributed electrical propulsion systems with higher aerodynamic efficiency and lower pollution emission have received extensive attention in recent years. The distributed electrical propulsion usually employs the new aero-propulsion integrated configuration. A simulation strategy for internal and external flow coupling based on the combination of lifting line theory and body force method is proposed. The lifting line theory and body force method as source term are embedded into the Navier-Stokes formulation. The lift and drag characteristics of the aero-propulsion coupling configuration are simulated. The results indicate that the coupling configuration has the most obvious lift augmentation at 12° angle of attack, which can provide an 11.11% increase in lift for the airfoil. At 0° angle of attack, the pressure difference on the lip parts provides the thrust component, which results in a lower drag coefficient. Additionally, the failure impact of a ducted fan at the middle or edge on aerodynamics is investigated. For the two failure conditions, the lift of the coupling configuration is decreased significantly by 27.85% and 26.14% respectively, and the lip thrust is decreased by 70.74% and 56.48% respectively.

Static hydroelastic study of composite T-foils with beam and lifting line models

Well-designed hydrofoils improve ship resistance and seakeeping by lifting the hull above the water. With greater speeds come greater loads, and the two-way interaction of structural deflections of lifting surfaces on the hydrodynamics must be considered. Tailored structural anisotropy can improve hydrodynamic and structural efficiency of lifting surfaces compared to rigid counterparts by exploiting the layup of composite materials. Structural efficiency here means reduced risk of structural failure for a given amount of material, and hydrodynamic efficiency means lower drag. A T-foil is a prototypical multi-component appendage, consisting of the foil wing and the strut, which we investigate in this work. We use simple composite beam and lifting line theory to explore the static fluid-structure interaction of a composite T-foil for a variety of fiber angles (θf = 0˝ , ˘15˝ ). We apply a simple approximation on lift coefficient at infinite Froude number (F n) to model the free-surface effects, which is valid at high depth-based Froude numbers (F nh ą10a h/c) when CL is independent of F nh and the inertial effects dominate. Results for the moth rudder T-foil geometry studied here indicate that aligning composite fibers towards the leading edge results in a more hydrostructurally efficient foil and that free-surface effects are minor because of the large submergence for this flow condition.

Minimum-Drag Fault-Tolerant Aircraft Control Allocation via Online Lifting Line Calculation

The minimization of drag at any given flight condition is necessary for the reduction of aircraft fuel consumption and is strongly linked to the way the different aerodynamic surfaces are deflected to control the flight trajectory. Current optimal control allocation methods calculate commands that minimize norm-based metrics that are only loosely related to aircraft drag. In this paper, using a novel real-time application of the lifting line concept, a new control allocation method for overactuated “biomorphic” fixed-wing aircraft is introduced, aiming at addressing the above limitation. The proposed technique outputs optimal, fault-tolerant minimum-drag control allocation solutions for vehicles with large numbers of aerodynamic surfaces, combined with angle-of-attack and angle-of-sideslip estimator functions that allow for direct, localized control of the lift force vectors. Owing to its close link to lifting line theory, which constitutes an integral part of the proposed allocation calculation, the method represents a low-computational-cost solution to the control allocation problem, easily adaptable to different aircraft configurations. Alongside its theoretical development and stability analysis, a series of simulated experiments are presented that demonstrate the proposed method’s characteristics and potential applications.

Combination of Advanced Actuator Line/Disk Model and High-Order Unstructured Finite Volume Solver for Helicopter Rotors

In the research field of rotorcraft aerodynamics, there are two fundamental challenges: resolving the complex vortex structures in rotor wakes and representing the moving rotor blades in the ambient airflow. In this paper, we address the first challenge by utilizing a third-order unstructured finite volume solver, which exhibits lower numerical dissipation than its second-order counterpart. This allows for sufficient resolution of small vortex structures on relatively coarse meshes. With this flow solver, the second challenge is addressed by modeling each rotor as an actuator disk (i.e., the actuator disk model (ADM)) or modeling each blade as an actuator line (i.e., the actuator line model (ALM)). Both of the two models are equipped with an improved tip loss correction, which is introduced in detail in the methodology section. In the section of numerical experiments, the numerical convergence properties of the two types of solvers have been compared in the case of two-dimensional infinite wing. In addition, the relationship between the ALM and the lifting line theory is discussed in the cases of fixed-wing calculations. Another goal of these cases is to validate the tip loss correction presented. The validation of the ALM/ADM and comparisons of computational efficiency are also demonstrated in simulations involving both hover and forward flight rotors. It was found that the combination of the third-order finite volume solver and the ALM/ADM with the improved tip loss correction presents an efficient way of performing the aerodynamic analysis of rotor-induced downwash flow.

An insight into the capability of the actuator line method to resolve tip vortices

Abstract. The actuator line method (ALM) is increasingly being preferred to the ubiquitous blade element momentum (BEM) approach in several applications related to wind turbine simulation, thanks to the higher level of fidelity required by the design and analysis of modern machines. Its capability to resolve blade tip vortices and their effect on the blade load profile is, however, still unsatisfactory, especially when compared to other medium-fidelity methodologies such as the lifting line theory (LLT). Despite the numerical strategies proposed so far to overcome this limitation, the reason for such behavior is still unclear. To investigate this aspect, the present study uses the ALM tool developed by the authors for the ANSYS® Fluent® solver (v. 20.2) to simulate a NACA0018 finite wing for different pitch angles. Three different test cases were considered: high-fidelity blade-resolved computational fluid dynamics (CFD) simulations (to be used as a benchmark), standard ALM, and ALM with the spanwise force distribution coming from blade-resolved data (frozen ALM). The last option was included to isolate the effect of force projection, using three different smearing functions. For the postprocessing of the results, two different techniques were applied: the LineAverage sampling of the local angle of attack along the blade and state-of-the-art vortex identification methods (VIMs) to outline the blade vortex system. The analysis showed that the ALM can account for tip effects without the need for additional corrections, provided that the correct angle of attack sampling and force projection strategies are adopted.

Quantifying the impact of modeling fidelity on different substructure concepts for floating offshore wind turbines – Part 1: Validation of the hydrodynamic module QBlade-Ocean

Abstract. To realize the projected increase in worldwide demand for floating offshore wind, numerical simulation tools must capture the relevant physics with a high level of detail while being numerically efficient. This allows engineers to have better designs based on more accurate predictions of the design driving loads, potentially enabling an economic breakthrough. The existing generation of offshore wind turbines is reaching a juncture, where traditional approaches, such as the blade element momentum theory, are becoming inadequate due to the increasing occurrence of substantial blade deflections. QBlade is a tool that includes a higher-fidelity aerodynamic model based on lifting-line theory, capable of accurately modeling such scenarios. In order to enable the simulation of offshore conditions in QBlade and to make use of this aerodynamic capability for novel offshore wind turbine designs, a hydrodynamic module called QBlade-Ocean was developed. In the present work, this module is validated and verified with two experimental campaigns and two state-of-the-art simulation frameworks on three distinct floating offshore wind turbine concepts. The results confirm the implementation work and fully verify QBlade as a tool to be applied in offshore wind turbine simulations. Moreover, a method aimed to improve the prediction of non-linear motions and loads under irregular wave excitation is analyzed in various conditions. This method results in a significant improvement in the surge and pitch degrees of freedom in irregular wave cases. Once wind loads are included, the method remains accurate in the pitch degree of freedom, while the improvements in the surge degree of freedom are reduced. A code-to-code comparison with the industry-designed Hexafloat concept highlights the coupled interactions on floating turbines that can lead to large differences in motion and load responses in otherwise identically behaving simulation frameworks.

Wing Planform Analysis with Taper Ratio and Expansion Segment Variation with Lifting Line Theory

Tapered wing shape of a planform wing is still widely used amongst airplane and UAVs with subsonic speed. In the design process, a good consideration for a good taper ratio of a wing is required to obtain the optimal and distribution for the desired function of an aircraft. Additionally, addition of expansion segment on wing planform shape is often used to increase the performance of wings without the increase of wingspan. Several methods to analyze a wing shape are experimentation, computational luid dynamics, and analytical calculation. Analyzing with analytical calculation will present limited, but accurate outcomes due to the assumptions that are made during the calculation. This method, however, is inexpensive. This is why analytical calculation is still a common method to use in the design process of an aircraft, particularly in the early phase. Five variants of taper ratio and 3 variants that with expansion segment is analyzed using the Lifting Line Theory that utilizes Fourier series at subsonic speed. The results are the values of and with respect to and the distribution of and along the wingspan. Increasing the taper ratio results in the decrease of and the increase of , while adding an expansion segment will give results that are dependent on the added segment’s taper ratio.

An efficient method for unsteady hydrofoil simulations, based on Non-Linear Dynamic Lifting Line theory

This paper presents a Non-linear Dynamic Lifting Line (NDLL)-based method for analysing hydrofoil vessels in waves. The method is validated, and its utility is demonstrated in a case study.The NDLL accounts for dynamic and 3D effects through a panellized wake and allows the evaluation of the pressure distribution on hydrofoils. Improved models are presented for wake deformation, wake composition, hydrofoil interaction, and the viscous core of vortex wake models. Furthermore, an efficient way of accounting for Green function terms through a matrix interpolation method is presented.Validation is performed by comparison with aerodynamic and hydrodynamic experiments and new 3D RANS simulations. The RANS simulations focus on geometries and operating conditions deemed representative of hydrofoiling fast ferries. Predictions of steady and dynamic lift, drag, wake velocities and cavitation inception are evaluated, and results correspond well with the benchmark data.A simulator framework is introduced, which integrates the hydrodynamic model with kinetics, kinematics, and control system models. This enables analyses of free-running vessels under active control, including assessments of resistance, motions, and the hydrofoil pressure distribution.The case study constitutes a qualitative verification of the simulator and flight control tuning methods, and reveals the possibility of negative added resistance in waves.

Generalized filtered lifting line theory for arbitrary chord lengths and application to wind turbine blades

AbstractThe filtered lifting line theory is an analytical approach used to solve the equations of flow subjected to body forces with a Gaussian distribution, such as used in the actuator line model. In the original formulation, the changes in chord length along the blade were assumed to be small. This assumption can lead to errors in the induced velocities predicted by the theory compared to full solutions of the equations. In this work, we revisit the original derivation and provide a more general formulation that can account for significant changes in chord along the blade. The revised formulation can be applied to wings with significant changes in chord along the span, such as wind turbine blades.

Experimental study on the mechanism of cavitation-induced ventilation

In this study, the cavitating flow and cavitation-induced ventilation flow around a surface-piercing hydrofoil were investigated to gain in-depth understanding of the interaction mechanism between the vaporous cavity and free surface at low cavitation numbers. Experiments were conducted in a constrained-launching water tank to visualize the cavity using a high-speed camera. Unsteady cloud cavitation and cavitation-induced ventilation at atmospheric pressure were observed and analyzed while piercing the free surface. The flow regime map was summarized at a fixed aspect ratio of ARh = 1.5. Subsequently, a physical model was proposed to predict the maximum depression depth of the water surface (H) at the trailing edge of the hydrofoil. Both the physical model and experimental results reveal that the non-dimensional depth H/c has a linear relation to Fn2 c × Rec × sin2α. Finally, a criterion for cavitation-induced ventilation based on the improved lifting-line theory and a physical model were proposed. A new relation H/Lc ∼ α0.5 was obtained, where Lc is the maximum cavity length. The results of this study can guide the design and application of hydrofoils for ventilation and cavitation processes.

Investigating the impact of various operating parameters on blade aeroelasticity and wake characteristics of large-scale wind turbines

The current study endeavors to investigate the impact of various operating parameters on the fluid-structure interaction characteristics of wind turbine blades and wake field characteristics within a wind farm. The observed underperformance of wind farms, attributed to downstream wind turbines being exposed to upstream wake effects, highlights the need for implementing active wake control (AWC). Consequently, in-depth research into the fluid-structure interaction characteristics of wind turbines and wake characteristics under AWC conditions becomes imperative. To address this, a novel aeroelastic analysis model based on geometrically exact beam theory, lifting line theory, and optimization is developed. Numerical simulations utilizing the IEA WIND 10 MW wind turbine, developed by the International Energy Agency, are employed to assess the fluid-structure interaction characteristics of the blades and wake characteristics under various wake control strategies. The results demonstrate that cone angle control in blade attitude alters the wake field width but may reduce velocity loss recovery rate, while pitch angle control significantly improves wake field length at the expense of increased response load. Yaw and tilt control redirect wake flow, with tilt control exhibiting lower thrust and structural response loads. The coupling of yaw and tilt control emerges as a promising approach with enhanced velocity loss recovery and reduced response loads. These findings contribute to understanding the effectiveness of wake control strategies in optimizing large-scale wind turbine performance.

Multidisciplinary design of aircraft wing for multiregime purposes

In this paper optimization process of the geometrical design of the wing is described. The goal is to minimize the aerodynamic drag at a requested lift in the cruise regime and thus reduce the fuel consumption. The second goal is to ensure the requested landing regime of the aircraft. Both cases are solved under the wing structural and weight considerations. As a fast design computational tool, a lifting line theory is used. It is moreover supplemented by non-linear aerodynamic airfoil characteristics to provide more accurate results of aerodynamic drag of the wing. These airfoil data were calculated with the CFD and enables to take significant accuracy between low and high-fidelity method which is verified in previous work. The list of requirements is set for aircraft for 40 passengers with a range of 2,500 km at a speed of 438 km/h. As the baseline the L-610 aircraft is selected. Optimized wing geometry in verified in cruise and landing regime/configuration with CFD solver. From the calculation of the weight of the wing and CFD results, the flight performance is calculated for the desired flight mission.

Actuator Disk Model with Improved Tip Loss Correction for Hover and Forward Flight Rotor Analysis

A novel actuator disk model (ADM) coupled with lifting-line theory is proposed in this paper. Several virtual planform blades are placed on a disk plane with a constant azimuthal interval, and the lifting-line theory is applied to each blade to predict the effective angle of attack. The proposed model considers the local lift and drag forces acting on disk surface cells by interpolating the predicted effective angle of attack with various azimuth angles to the actuator disk plane; therefore, the proposed model considers individual blade tip vortices without tip loss functions. Experimental data for hover and forward flight rotors are used to validate the proposed model. For hovering flight, sectional thrust based on collective pitch angles predicted by the modified ADM was similar to that obtained in the experiments. For forward flight, the inflow above the rotor estimated by the proposed ADM was similar to that obtained in the experiments and by using other numerical methods. Thus, the developed ADM can be used for rotor performance analysis under the main flight conditions of V/STOL.

Experimental analysis of the wake behind a small wind-turbine model in yaw

In this work we study the wake of a yawed wind-turbine model immersed in an atmospheric boundary layer (ABL). The ABL is replicated in the wind tunnel by means of a barrier-spires and distributed roughness configuration and is representative of a rural terrain. We quantify the properties of the wake in the horizontal plane at hub height and compare the predictions of available wake models to our data for different yaw angles. It is found that the model based on lifting-line theory performs best in predicting the velocity deficit without the need of tuning the parameters to the current setup. However, the wake deflection is slightly underestimated, most notably at the transition between near and far wake. Furthermore, a comparison with the turbine in a uniform incoming flow highlights the enhanced downward deflection of the wake which results from its interaction with the ABL.

Verification &amp; validation of lifting line - and -formulations for 3-D planforms under viscous flows

AbstractMany adaptations of the lifting-line theory have been developed since its conception to aid in preliminary aerodynamic wing design, but they typically fall into two main formulations, named $\alpha $ - and $\Gamma $ -formulation, which differ in terms of the control points chordwise location and the variable updated during the iterative scheme. This paper assess the advantages and drawbacks of both formulations through the implementation of the respective methods and application of standard verification and validation procedures. Verification showed that the $\Gamma $ -method poorly converges for wings with nonstraight quarter-chord lines, while the $\alpha $ -method presents adequate convergence rates and uncertainties for all geometries; it also showed that the $\Gamma $ -method agrees best with analytic results from the cassic lifting-line theory, indicating that it tends to overpredict wing lift. Validation and comparison to other modern lifting-line methods was done for similar geometries, and not only corroborated the poor converge and lift overprediction of the $\Gamma $ -method, but also showed that the $\alpha $ -method presented the closest results to experimental data for almost all cases tested, concluding that this formulation is typically superior regardless of the wing geometry. These results indicate that the implemented $\alpha $ -method has a greater potential for the extension of the lifting-line theory to more geometrically complex lifting surfaces other than fixed wings with straight quarter-chord lines and wakes constrained to the planform plane.

물리 기반 자유후류 모델링을 통한 소형 무인기용 동축 로터 블레이드의 최적 설계

Coaxial rotors can generate high thrust with a small aircraft volume, which makes them an attractive configuration for small unmanned aerial vehicles (UAV), particularly for hovering and low-speed flights. However, the aerodynamic interference between the upper and lower rotors makes it more challenging to predict design trends for aerodynamic performance. Three-dimensional compressible unsteady Reynolds-averaged Navier-Stokes (3D URANS) solver can be a suitable choice, however, it requires burdensome computational cost for design optimization. One alternative that can alleviate the computational burden of design optimization with little loss of accuracy is lifting-line theory coupled with a free-wake method. Although the free-wake method provides a high degree of freedom in modeling the wake effects, a well-adjusted wake model can provide an accurate prediction of aerodynamic performance and design trends. In this study, the free-wake model has been adjusted to correlate the aerodynamic environment, such as sectional thrust, induced velocity, the effective angle of attack, and wake geometry, predicted by the 3D URANS simulation for the baseline coaxial rotor. By adjusting the physical model parameters of free-wake, such as core type, growth rate, and its initial strength, the free-wake model is adjusted based on the physical relationship, not the simple linear scaling factor generally used to inflow factor. The optimal coaxial rotor was derived based on the performance prediction of the adjusted free-wake model. The 3D URANS simulation was also conducted for the optimal coaxial rotor and compared the aerodynamic environments predicted by the adjusted freewake model. The performance index and aerodynamic environments predicted by both solvers also showed similar results for the optimal coaxial rotor. The present physics-based freewake adjustment can be an efficient design approach with consideration of the complex aerodynamic phenomena of a coaxial rotor system.