How should the lift and drag forces be calculated from 2-D airfoil data for dihedral or coned wind turbine blades?

In the present work, a consistent method for calculating the lift and drag forces from the 2-D airfoil data for the dihedral or coned horizontal-axis wind turbines when using generalized lifting-line methods is described. The generalized lifting-line methods include, for example, lifting-line (LL), actuator line (AL), blade element momentum (BEM) and blade element vortex cylinder (BEVC) methods. A consistent interpretation of classic unsteady 2-D thin airfoil theory results for use in a generally moving frame of reference within a linearly varying onset velocity field reveals that it is necessary to use not only the relative flow magnitude and direction at one point along the chord line (for instance three-quarter-chord), but also the gradient of the flow direction in the chordwise direction (or, equivalently, the flow direction at the quarter-chord) to correctly determine the magnitude and direction of the resulting 2-D aerodynamic forces and moment. However, this aspect is generally overlooked and most implementations in generalized lifting-line methods use only the flow information at one calculation point per section for simplicity. This simplification will not change the performance prediction of planar rotors, but will cause an error when applied to non-planar rotors. The present work proposes a generalized method to correct the error introduced by this simplified single-point calculation method. In this work this effect is investigated using the special case, where the wind turbine blade has only dihedral and no sweep, operating at steady-state conditions with uniform inflow applied perpendicular to the rotor plane. We investigate the impact of the effect by comparing the predictions of the steady-state performance of non-planar rotors from the consistent approach with the simplified one-point approach of the LL method. The results are verified using blade geometry resolving Reynolds-averaged Navier-Stokes (RANS) simulations. The numerical investigations confirmed that the correction derived from thin airfoil theory is needed for the calculations to correctly determine the magnitude and direction of the sectional aerodynamic forces for non-planar rotors. The aerodynamic loads of upwind and downwind coned blades that are calculated using the LL method, the BEM method, the BEVC method and the AL method are compared for the simplified and the full method. Results using the full method, including different specific implementation schemes, are shown to agree significantly better with fully-resolved RANS than the often used simplified one-point approaches.

Horizontal Axis Wind Turbine Performance Analysis

The present work uses the method of Blade Element Momentum Theory as suggested by Hansen. The method applied to three blade models adopted from Rahgozar S. with the airfoil data used the data provided by Wood D. The wind turbine performance described in term of the thrust coefficient C_T, torque coefficient C_Q and the power coefficient C_p . These three coefficient can be deduced from the Momentum theory or from the Blade element Theory(BET). The present work found the performance coefficient derived from the Momentum theory tent to over estimate. It is suggested to used the BET formulation in presenting these three coefficients. In overall the Blade Element Momentum Theory follows the step by step as described by Hansen work well for these three blade models. However a little adjustment on the blade data is needed. To the case of two bladed horizontal axis wind

A comprehensive comparative investigation of the Lifting Line Theory and Blade Element Momentum Theory applied to small wind turbines

Small wind turbines are adequate for electricity generation in isolated areas to promote local expansion of commercial activities and social inclusion. Blade element momentum (BEM) method is usually used for performance prediction, but generally produces overestimated predictions since the wake effects are not precisely accounted for. Lifting line theory (LLT) can represent the blade and wake effects more precisely. In the present investigation the two methods are analyzed and their predictions of the aerodynamic performance of small wind turbines are compared. Conducted simulations showed a computational time of about 149.32 s for the Gottingen GO 398 based rotor simulated by the BEM and 1007.7 s for simulation by the LLT. The analysis of the power coefficient showed a maximum difference between the predictions of the two methods of about 4.4% in the case of Gottingen GO 398 airfoil based rotor and 6.3% for simulations of the Joukowski J 0021 airfoil. In the case of the annual energy production a difference of 2.35% is found between the predictions of the two methods. The effects of the blade geometrical variants such as twist angle and chord distributions increase the numerical deviations between the two methods due to the big number of iterations in the case of LLT. The cases analyzed showed deviations between 3.4% and 4.1%. As a whole, the results showed good performance of both methods; however the lifting line theory provides more precise results and more information on the local flow over the rotor blades.

Wake Expansion and the Finite Blade Functions for Horizontal-Axis Wind Turbines

This paper considers the effect of wake expansion on the finite blade functions in blade element/momentum theory for horizontal-axis wind turbines. For any velocity component, the function is the ratio of the streamtube average to that at the blade elements. In most cases, the functions are set by the trailing vorticity only and Prandtl’s tip loss factor can be a reasonable approximation to the axial and circumferential functions at sufficiently high tip speed ratio. Nevertheless, important cases like coned or swept rotors or shrouded turbines involve more complex blade functions than provided by the tip loss factor or its recent modifications. Even in the presence of significant wake expansion, the functions derived from the exact solution for the flow due to constant pitch and radius helical vortices provide accurate estimates for the axial and circumferential blade functions. Modifying the vortex pitch in response to the expansion improves the accuracy of the latter. The modified functions are more accurate than the tip loss factor for the test cases at high tip speed ratio that are studied here. The radial velocity is important for expanding flow as it has the magnitude of the induced axial velocity near the edge of the rotor. It is shown that the resulting angle of the flow to the axial direction is small even with significant expansion, as long is the tip speed ratio is high. This means that blade element theory does not have account for the effective blade sweep due to the radial velocity. Further, the circumferential variation of the radial velocity is lower than of the other components.

Hydrodynamics modeling of a piezoelectric micro-robotic fish with double caudal fins

An analytical hydrodynamics model for a piezoelectric micro-robotic fish with double caudal fins is presented in this paper. The relation between displacement of the piezoelectric actuator and oscillating angle of the caudal fin is established based on the analysis of the flexible four-bar linkage transmission. The hydrodynamics of caudal fins are described by airfoil and blade element theories. Furthermore, the dynamics and kinetics of the whole micro-robotic fish are analyzed and validated by experiments.

Surrogate models for the blade element momentum aerodynamic model using non-intrusive Polynomial Chaos Expansions

In typical industrial practice based on IEC standards, wind turbine simulations are computed in the time domain for each mean wind speed bin using a few number of unsteady wind seeds. Software such as FAST, BLADED or HAWC2 can be used to capture the unsteadiness and uncertainties of the wind in the simulations. The statistics of these aeroelastic simulations output are extracted and used to calculate fatigue and extreme loads on the wind turbine components. The minimum requirement of having six seeds does not guarantee an accurate estimation of the overall statistics. One solution might be running more seeds; however, this will increase the computation cost. Moreover, to move beyond Blade Element Momentum (BEM) based tools toward vortex/potential flow formulations, a reduction in the computational cost associated with the unsteady flow and uncertainty handling is required. This study illustrates the unsteady wind aerodynamic statistics' stationary character based on the standard turbulence models. This character is shown based on the output of NREL 5MW simulations. Afterwards, we propose a non-intrusive Polynomial Chaos Expansion approach to build a surrogate model of the loads' statistics, NREL 5MW rotor thrust and torque, at each time step, to estimate the extreme statistics more accurately and efficiently.

Wind Tunnel Tests of a Model Small-scale Horizontal-axis Wind Turbine Developed from Blade Element Momentum Theory

The small-scale horizontal-axis wind turbines (SHAWTs) have emerged as the promising alternative energy resource for the off-grid electrical power generation. These turbines primarily operate at low Reynolds number, low wind speed, and low tip speed ratio conditions. Under such circumstances, the airfoil selection and blade design of a SHAWT becomes a challenging task. The present work puts forward the necessary steps starting from the aerofoil selection to the blade design and analysis by means of blade element momentum theory (BEMT) for the development of four model rotors composed of E216, SG6043, NACA63415, and NACA0012 airfoils. This analysis shows the superior performance of the model rotor with E216 airfoil in comparison to other three models. However, the subsequent wind tunnel study with the E216 model, a marginal drop in its performance due to mechanical losses has been observed.

Programa de Cómputo para Estimar el Rendimiento de una Hélice Empleando una Corrección por Número de Mach a la Teoría Combinada

En este trabajo se propone una corrección empírica por número de Mach a la teoría combinada para hélices y se describe el programa de cómputo desarrollado para determinar el comportamiento de la misma. El programa requiere como datos de entrada la geometría de la hélice y los coeficientes aerodinámicos en función del número de Mach de los perfiles de la pala de la hélice. Éste calcula los coeficientes aerodinámicos y las velocidades inducidas de cada elemento de pala empleando la teoría combinada, corrige los coeficientes aerodinámicos por efecto de compresibilidad y calcula la eficiencia, así como los coeficientes de tracción y de potencia de la hélice para diferentes velocidades de avance y, finalmente, los presenta en forma gráfica. Se observa que los resultados obtenidos con la teoría combinada corregida por número de Mach fueron satisfactorios ya que se aproximan más a los resultados experimentales que la teoría combinada simple. This work proposes an empirical correction by Mach number to the BEM (Blade-Element Momentum) Theory for propellers and describes the software developed to determine the behavior of it. The input for the software is the geometry of the propeller and the aerodynamic coefficient in function of the Mach number for the airfoils used for the propeller chosen. The software calculates the aerodynamic coefficients and the induced velocities at each station of the blade of the propeller using the BEM theory, then corrects these coefficients by the effect of compressibility and calculates the efficiency, the traction and power coefficients for a range of forward velocities, and finally presents a graph with the results obtained. We can observe that the results obtain are satisfactory comparing with the experimental results and obtaining lower difference error by this method than with the simple BEM theory.