Numerical investigation of wind turbine wake development in directionally sheared inflow

Turbines in wind farms are subject to complex mutual aerodynamic interactions, which in detail depend upon the characteristics of the atmospheric boundary layer. Our two objectives with this paper were to investigate the impact of directionally sheared inflow on the wake development behind a single wind turbine and to analyse the impact of the wakes on the energy yield and loading of a downstream turbine, which is exposed to partial and full wake conditions. We performed simulations with a framework based on a coupled approach of large‐eddy simulation and an actuator line representation of an aeroelastic turbine model. Our results show that directionally sheared inflow leads to a non‐symmetrical wake development, which transfers to distinct differences in the energy yield and loading of downstream turbines of equal lateral offsets in opposite direction. Therefore, the assumption of wakes being axisymmetrical could lead to notable deviations in the prediction of wake behaviour and their impact on downstream turbines for atmospheric inflow conditions, which include directional shear. Copyright © 2016 John Wiley & Sons, Ltd.     

Performance and wake of a Savonius vertical‐axis wind turbine under different incoming conditions

In this study, the performance, drag, and horizontal midplane wake characteristics of a vertical‐axis Savonius wind turbine are investigated experimentally. The turbine is drag driven and has a helical configuration, with the top rotated 180° relative to the bottom. Both performance and wake measurements were conducted in four different inflow conditions, using Reynolds numbers of Re_D≈1.6×10⁵ and Re_D≈2.7×10⁵ and turbulence intensities of 0.6% and 5.7%. The efficiency of the turbine was found to be highly dependent on the Reynolds number of the incoming flow. In the high Reynolds number flow case, the efficiency was shown to be considerably higher, compared with the lower Reynolds number case. Increasing the incoming turbulence intensity was found to mitigate the Reynolds number effects. The drag of the turbine was shown to be independent of the turbine's rotational speed over the range tested, and it was slightly lower when the inflow turbulence was increased. The wake was captured for the described inflow conditions in both optimal and suboptimal operating conditions by varying the rotational speed of the turbine. The wake was found to be asymmetrical and deflected to the side where the blade moves opposite to the wind. The largest region of high turbulent kinetic energy was on the side where the blade is moving in the same direction as the wind. Based on the findings from the wake measurements, some recommendations on where to place supplementary turbines are made.     

Wind tunnel investigation on the effect of the turbine tower on wind turbines wake symmetry

In wind farms, the wake of the upstream turbines becomes the inflow for the downstream machines. Ideally, the turbine wake is a stable vortex system. In reality, because of factors like background turbulence, mean flow shear, and tower‐wake interaction, the wake velocity deficit is not symmetric and is displaced away from its mean position. The irregular velocity profile leads to a decreased efficiency and increased blade stress levels for the downstream turbines. The object of this work is the experimental investigation of the effect of the wind turbine tower on the symmetry and displacement of the wake velocity deficit induced by one and two in‐line model wind turbines (,D= 0.9 m). The results of the experiments, performed in the closed‐loop wind tunnel of the Norwegian University of Science and Technology in Trondheim (Norway), showed that the wake of the single turbine expanded more in the horizontal direction (side‐wall normal) than in the vertical (floor normal) direction and that the center of the wake vortex had a tendency to move toward the wind tunnel floor as it was advected downstream from the rotor. The wake of the turbine tandem showed a similar behavior, with a larger degree of non‐symmetry. The analysis of the cross‐stream velocity profiles revealed that the non‐symmetries were caused by a different cross‐stream momentum transport in the top‐tip and bottom‐tip region, induced by the turbine tower wake. In fact, when a second additional turbine tower, mirroring the original one, was installed above the turbine nacelle, the wake recovered its symmetric structure. Copyright © 2017 John Wiley & Sons, Ltd.     

Flow characterization in the near‐wake region of a horizontal axis wind turbine

An experimental study is conducted to investigate the flow dynamics within the near‐wake region of a horizontal axis wind turbine using particle image velocimetry (PIV). Measurements were performed in the horizontal plane in a row of four radially distributed measurement windows (tiles), which are then patched together to obtain larger measurement field. The mean and turbulent components of the flow field were measured at various blade phase angles. The mean velocity and turbulence characteristics show high dependency on the blade phase angle in the near‐wake region closer to the blade tip and become phase independent further downstream at a distance of about one rotor diameter. In the near‐wake region, both the mean and turbulent characteristics show a systemic variation with the phase angle in the blade tip region, where the highest levels of turbulence are observed. The streamlines of the instantaneous velocity field at a given phase allowed to track a tip vortex which showed wandering trend. The tip vortices are mostly formed at r/R > 1, which indicates the wake expansion. Results also show the gradual movement of the vortex region in the axial direction, which can be attributed to the dynamics of the helical tip vortices which after being generated from the tip, rotate with respect to the blade and move in the axial direction because of the axial momentum of the flow. The axial velocity deficit was compared with other laboratory and field measurements. The comparison shows qualitative similarity. Copyright © 2015 John Wiley & Sons, Ltd.     

基于流固耦合偏航下风力机尾迹湍流特征的研究

偏航状态下风力机叶片与流场之间相互作用会导致风力机近尾迹流场的湍流特征变化,采用双向流固耦合对不同偏航工况下水平轴风力机近尾迹流场进行数值模拟研究,获得不同偏航角下尾迹湍流特征演化规律。结果表明：随着偏航角的增大,正偏航侧会出现“速度亏损圆环”,且此圆环的范围呈扩大趋势;偏航角的增大对叶根处速度亏损影响最大,对叶尖处速度亏损影响最小,与正偏航侧相比,负偏航侧的速度亏损值减为约1/2;随着偏航角的增大,正负偏航侧的湍流强度变化呈不对称性,正偏航侧对湍流耗散的影响程度较负偏航侧大;涡流黏度越来越小,且在偏航10°涡流黏度相对于偏航5°减小约1/2,沿着轴向叶尖涡的管状环涡结构变得不稳定,出现明显耗散,且在偏航15°之后涡结构的耗散破裂程度越来越剧烈,进而对风力机气动噪声产生较大影响。     

The spectral signature of wind turbine wake meandering: A wind tunnel and field‐scale study

Field‐scale and wind tunnel experiments were conducted in the 2D to 6D turbine wake region to investigate the effect of geometric and Reynolds number scaling on wake meandering. Five field deployments took place: 4 in the wake of a single 2.5‐MW wind turbine and 1 at a wind farm with numerous 2‐MW turbines. The experiments occurred under near‐neutral thermal conditions. Ground‐based lidar was used to measure wake velocities, and a vertical array of met‐mounted sonic anemometers were used to characterize inflow conditions. Laboratory tests were conducted in an atmospheric boundary layer wind tunnel for comparison with the field results. Treatment of the low‐resolution lidar measurements is discussed, including an empirical correction to velocity spectra using colocated lidar and sonic anemometer. Spectral analysis on the laboratory‐ and utility‐scale measurements confirms a meandering frequency that scales with the Strouhal number St = fD/U based on the turbine rotor diameter D. The scaling indicates the importance of the rotor‐scaled annular shear layer to the dynamics of meandering at the field scale, which is consistent with findings of previous wind tunnel and computational studies. The field and tunnel spectra also reveal a deficit in large‐scale turbulent energy, signaling a sheltering effect of the turbine, which blocks or deflects the largest flow scales of the incoming flow. Two different mechanisms for wake meandering—large scales of the incoming flow and shear instabilities at relatively smaller scales—are discussed and inferred to be related to the turbulent kinetic energy excess and deficit observed in the wake velocity spectra.     

Investigation of the theoretical load alleviation potential using trailing edge flaps controlled by inflow data

A novel control concept for fatigue load reduction with trailing edge flaps based on the measurement of the inflow locally on the blade was presented. The investigation was conducted with the aeroelastic code HAWC2. The aerodynamic modelling in the code is based on blade element momentum theory. The simulations were carried out for the NREL 5MW reference wind turbine, and the mean wind speed at hub height was 8 m s^?1. The turbine was operated with fixed rotational speed. The energy at the blade is concentrated in spectral bands centred at multiples of the rotational frequency up to three times the rotational frequency. The highest fatigue load reduction was achieved when the inflow sensor was placed at the outer parts of the blade. In the best case, the reduction of the local fatigue loads induced by the blade sectional normal force was 60%. The control method gave the highest fatigue load reductions in conditions with strong wind shear. The demands for the flap actuator in terms of deflection angles was ±10°. The requirements in terms of the flap deflection velocity depend mainly on the inflow turbulence intensity. The maximum value was ±40°s^?1 for 20% inflow turbulence intensity. Unsteady aerodynamic effects seem to be negligible. Copyright © 2015 John Wiley & Sons, Ltd.     

大粗糙度弦向分布对风力机气动性能影响

针对5 mm大尺度蚊虫尸体在叶片上的附积问题,采用带转捩的k-ω SST湍流模型,以NERL Phase VI风力机为研究对象,从不同弦向覆盖位置入手,对大粗糙度下风力机气动性能进行数值模拟。研究结果表明,粗糙度对风力机整体做功具有较大影响,粗糙度越大效率降低越显著;粗糙度使在叶片靠近叶尖位置的吸力面促成小型低压涡,转捩提前;该效应在大尺度粗糙条件下表现明显,附着涡强度也更大;在叶片压力面添加75%c粗糙度会使翼型产生的气动损失最大。     

Vortex‐induced vibrations on a modern wind turbine blade

This article investigates the aero‐elastic response of the DTU 10‐MW RWT blade in deep stall conditions with angles of attack in the vicinity of 90 degrees. The simulations were conducted with the high‐fidelity fluid–structure interaction simulation tool HAWC2CFD employing the multi‐body‐based structural model of HAWC2 and the incompressible computational fluid dynamics solver EllipSys3D. The study utilizes detached eddy simulation computations and considers the three‐dimensional blade geometry including blade twist and taper. A preliminary frequency analysis of the load variations on a stiff blade showed that an inclined inflow with a velocity component along the blade axis can trigger a spanwise correlated vortex shedding over large parts of the blade. Moderate wind speeds were sufficient to generate vortex shedding with frequencies close to the first edgewise eigenfrequency of the blade. Aero‐elastic computations of the elastic blade confirmed the findings of the frequency analysis. Inflow conditions with inclination angles between Ψ = 20° and Ψ = 55° and relatively low to moderate wind speeds between V = 16 and V = 26ms^?1 were sufficient to trigger severe edgewise blade vibrations with blade tip amplitudes of several metres. The investigated inflow conditions are considered realistic and might occur when the wind turbine is idling or standing still and the yaw system is unable to align the wind turbine with the incoming wind. Copyright © 2016 John Wiley & Sons, Ltd.     

An experimental study on the effects of winglets on the tip vortex interaction in the near wake of a model wind turbine

An experimental study of the near wake up to four rotor diameters behind a model wind turbine rotor with two different wing tip configurations is performed. A straight‐cut wing tip and a downstream‐facing winglet shape are compared on the same two‐bladed rotor operated at its design tip speed ratio. Phase‐averaged measurements of the velocity vector are synchronized with the rotor position, visualizing the downstream location of tip vortex interaction for the two blade tip configurations. The mean streamwise velocity is found not to be strongly affected by the presence of winglet tip extensions, suggesting an insignificant effect of winglets on the time‐averaged inflow conditions of a possible downstream wind turbine. An analysis of the phase‐averaged vorticity, however, reveals a significantly earlier tip vortex interaction and breakup for the wingletted rotor. In contradistinction, the tip vortices formed behind the reference configuration are assessed to be more stable and start merging into larger turbulent structures significantly further downstream. These results indicate that an optimized winglet design can not only contribute to a higher energy extraction in a rotor's tip region but also can positively affect the wake's mean kinetic energy recovery by stimulating a faster tip vortex interaction.     

Lift correction model for local shear flow effect on wind turbine airfoils

Wind turbines operate under various wind conditions in which turbulence virtually always exists. Therefore, unsteady wind turbine simulation methods to estimate wind loading in turbulent inflow conditions are very important for developing optimally designed wind turbines. Several methods have been developed for this purpose and are usually based on the blade element momentum theory (BEMT), which is used for calculation of the wind loading on turbine blades. The local shear flow effect induced by turbulence, however, is not explicitly considered in the popular BEMT-based simulations. Extreme situations can occur in a large-scale wind farm where the inflow field of a wind turbine may contain strong tip vortices generated from upstream turbines. In this study, the effects of idealized local shear flows around a two-dimensional airfoil, S809, on its aerodynamic characteristics were analyzed by CFD simulations. Various parameters including reference inflow velocity, shear rate, angle of attack, and cord length of the airfoil were examined. From the simulation results, several important characteristics were found. The shear rate in a flow causes some changes in the lift coefficient depending on its sign and magnitude, while the angle of attack does not have a distinguishable influence. The chord length and reference inflow also cause proportional and inversely proportional changes in the lift coefficient, respectively. Based on these observations, we adopted an analytic expression for the lift coefficient from the thin airfoil theory and proposed a lift correction model, which is easily applicable to the traditional load analysis procedure based on the BEMT.     

Influence of wake dynamics on the performance and aeroelasticity of wind turbines

Wind turbines are currently a rapidly expanding form of renewable energy. However, there are numerous technological challenges that must be overcome before wind energy provides a significant amount of power in the United States. One of the primary challenges in wind turbine design and analysis is accurately accounting for the aerodynamic environment. This study is focused on a comprehensive verification and validation of the NREL FAST code, which is enhanced to include a free vortex wake model. The verification and validation is carried out through a comparison of blade lift distribution, wind turbine power and force and moment coefficients using a combination of CFD and experimental data. The results are also compared against Blade Element Momentum theory, and results from a 2001 double-blind NREL study on the prediction capabilities of wind turbine modeling tools. Results indicate that the enhanced aeroelastic code generally provides improved predictions. However, in several notable cases the predictions are only marginally improved, or even worse, than those generated using Blade Element Momentum theory aerodynamics. It is concluded that modeling of the aerodynamic environment remains incomplete, even after inclusion of wake effects. One important aspect identified is modeling of the unsteady aerodynamic lift characteristics of the rotor. Finally, the aeroelastic response in the combined presence of wake effects and inflow turbulence is examined. Significant differences are observed in loads, power, and structural response between results computed using the free wake model or simpler models, such as Blade Element Momentum theory.     

Wind turbine response to parameter variation of analytic inflow vortices

As larger wind turbines are placed on taller towers, rotors frequently operate in atmospheric conditions that support organized, coherent turbulent structures. It is hypothesized that these structures have a detrimental impact on the blade fatigue life experienced by the wind turbine. These structures are extremely difficult to identify with sophisticated anemometry such as ultrasonic anemometers. This study was performed to identify the vortex characteristics that contribute to high‐amplitude cyclic blade loads, assuming that these vortices exist under certain atmospheric conditions. This study does not attempt to demonstrate the existence of these coherent turbulent structures. In order to ascertain the idealized worst‐case scenario for vortical inflow structures impinging on a wind turbine rotor, we created a simple, analytic vortex model. The Rankine vortex model assumes that the vortex core undergoes solid body rotation to avoid a singularity at the vortex centre and is surrounded by a two‐dimensional potential flow field. Using the wind turbine as a sensor and the FAST wind turbine dynamics code with limited degrees of freedom, we determined the aerodynamic loads imparted to the wind turbine by the vortex structure. We varied the size, strength, rotational direction, plane of rotation, and location of the vortex over a wide range of operating parameters. We identified the vortex conformation with the most significant effect on the blade root bending moment cyclic amplitude. Vortices with radii on the scale of the rotor diameter or smaller caused blade root bending moment cyclic amplitudes that contribute to high damage density. The rotational orientation, clockwise or counter‐clockwise, produces little difference in the bending moment response. Vortices in the XZ plane produce bending moment amplitudes significantly greater than vortices in the YZ plane. Published in 2005 by John Wiley & Sons, Ltd.     

Wind turbine performance in shear flow and in the wake of another turbine through high fidelity numerical simulations with moving mesh technique

We present numerical simulations of two horizontal axis wind turbines, one operating under the wake of the other, under an incoming sheared velocity profile. We use a moving mesh technique to represent the rotation of the turbine blades and solve the unsteady Reynolds averaged Navier–Stokes equations with a shear stress transport k ? ω turbulence model. Temporal evolution of the lift and drag coefficients of the front turbine show a phase shift in the periodic cycle due to the non‐uniform incoming free stream velocity. Comparisons of the lift and drag coefficients for the back turbine with the unperturbed behaviour of the front demonstrate the complex non‐linear interactions of the blades with the wake, with a significant decrease in overall performance and two peaks at specific points in the cycle associated with local angle of attack modification in the wake. The vorticity field in the near wake shows tilting of the vortex lines in the wake due to the shear and a faster diffusion of the tip vortical signature compared with the uniform free stream velocity case. Observations of the wake–wake interaction show good agreement with recent studies that use different methodologies. Copyright © 2012 John Wiley & Sons, Ltd.     

Analysis of vortex‐induced and stall‐induced vibrations at standstill conditions using a free wake aerodynamic code

Vortex‐induced and stall‐induced vibrations of a 2D elastically mounted airfoil at high angles of attack in the vicinity of 90° are investigated using a vortex type model. Such conditions are encountered in parked or idling operation at extreme yaw angles provoked by control system failures. At very high angles of attack, massive flow separation takes place over the entire blade span, and vortex shedding evolves downstream of the blade giving rise to periodically varying loads at frequencies corresponding to the Strouhal number of the vortices shed in the wake. As a result, vortex‐induced vibrations may occur when the shedding frequency matches the natural frequency of the blade. A vortex type model formulated on the basis of the ‘double wake’ concept is employed for the modelling of the stalled flow past a 2D airfoil. By tuning the core size of the vortex particles in the wake, the model predictions are successfully validated against averaged 2D measurements on a DU‐96‐W‐180 airfoil at high angles of attack. In order to assess the energy fed to the airfoil by the aerodynamic loads, the behaviour under imposed sinusoidal edgewise motions is analysed for various oscillation frequencies and amplitudes. Moreover, stall‐induced and vortex‐induced vibrations of an elastically mounted airfoil section are assessed. The vortex model predicts higher aeroelastic damping as compared with that obtained using steady‐state aerodynamics. Excessive combined vortex‐induced and stall‐induced edgewise vibrations are obtained beyond the wind speed of 30 m s^?1. Copyright © 2014 John Wiley & Sons, Ltd.     

Large‐eddy simulation of stable boundary layer turbulence and estimation of associated wind turbine loads

Stochastic simulation of turbulent inflow fields commonly used in wind turbine load computations is unable to account for contrasting states of atmospheric stability. Flow fields in the stable boundary layer, for instance, have characteristics such as enhanced wind speed and directional shear; these effects can influence loads on utility‐scale wind turbines. To investigate these influences, we use large‐eddy simulation (LES) to generate an extensive database of high‐resolution ( ～ 10 m), four‐dimensional turbulent flow fields. Key atmospheric conditions (e.g., geostrophic wind) and surface conditions (e.g., aerodynamic roughness length) are systematically varied to generate a diverse range of physically realizable atmospheric stabilities. We show that turbine‐scale variables (e.g., hub height wind speed, standard deviation of the longitudinal wind speed, wind speed shear, wind directional shear and Richardson number) are strongly interrelated. Thus, we strongly advocate that these variables should not be prescribed as independent degrees of freedom in any synthetic turbulent inflow generator but rather that any turbulence generation procedure should be able to bring about realistic sets of such physically realizable sets of turbine‐scale flow variables. We demonstrate the utility of our LES‐generated database in estimation of loads on a 5‐MW wind turbine model. More importantly, we identify specific turbine‐scale flow variables that are responsible for large turbine loads—e.g., wind speed shear is found to have a greater influence on out‐of‐plane blade bending moments for the turbine studied compared with its influence on other loads such as the tower‐top yaw moment and the fore‐aft tower base moment. Overall, our study suggests that LES may be effectively used to model inflow fields, to study characteristics of flow fields under various atmospheric stability conditions and to assess turbine loads for conditions that are not typically examined in design standards. Copyright © 2013 John Wiley & Sons, Ltd.     

Effects of inflow turbulence intensity and turbine arrangements on the power generation efficiency of large wind farms

In this study, we conduct a series of large‐eddy simulations (LESs) to study the impact of different incoming turbulent boundary layer flows over large wind farms, with a particular focus on the overall efficiency of electricity production and the evolution of the turbine wake structure. Five representative turbine placements in the large wind farm are considered, including an aligned layout and four staggered layouts with lateral or vertical offset arrangements. Four incoming flow conditions are used and arranged from the LESs of the ABL flow over homogeneous flat surfaces with four different aerodynamic roughness lengths (i.e., z0 = 0.5, 0.1, 0.01, and 0.0001 m), where the hub‐height turbulence intensity levels are about 11.1%, 8.9%, 6.8%, and 4.9%, respectively. The simulation results indicate that an enhancement in the inflow turbulence level can effectively increase the power generation efficiency in the large wind farms, with about 23.3% increment on the overall farm power production and up to about 32.0% increment on the downstream turbine power production. Under the same inflow condition, the change of the turbine‐array layouts can increase power outputs within the first 10 turbine rows, which has a maximum increment of about 26.5% under the inflow condition with low turbulence. By comparison, the increase of the inflow turbulence intensity facilitates faster wake recovery that raises the power generation efficiency of large wind farms than the adjustment of the turbine placing layouts.     

Blade load mitigation control design for a wind turbine operating in the path of vortices

As wind turbine rotor size continues to increase, load mitigation becomes an important control objective. Turbines with hub heights of nearly 100m operate in the stable, nocturnal boundary layer where coherent turbulence can be generated by atmospheric phenomena outside the surface layer. These coherent turbulent structures may contribute to blade fatigue loads that can be mitigated with advanced control algorithms. Disturbance accommodating control (DAC) methods were implemented in a wind turbine structural dynamics simulation code to mitigate transient blade load response induced by a simple, Rankine vortex in the inflow. As a best‐case scenario, a full‐state feedback controller (which included a very detailed disturbance model) showed that blade flap damage equivalent load caused by the vortex passing through the rotor could be reduced by 30% compared to one that resulted from simulation of a typical proportional‐integral (PI) controller. A realizable DAC controller that incorporates only the vertical shear component of the vortex reduced loads by 9% compared to that resulting from simulation of a PI controller. The load reduction was even greater when the vortex was superimposed over full‐field, homogeneous turbulence. DAC methods have the flexibility to incorporate properties of coherent turbulent inflow structures in the controller design to mitigate blade fatigue loads. Further work must be done to develop disturbance models as more details about the turbulent structures are identified. Copyright © 2007 John Wiley & Sons, Ltd.     

Reynolds number dependence of turbulence statistics in the wake of wind turbines

A wind tunnel experiment has been performed to quantify the Reynolds number dependence of turbulence statistics in the wake of a model wind turbine. A wind turbine was placed in a boundary layer flow developed over a smooth surface under thermally neutral conditions. Experiments considered Reynolds numbers on the basis of the turbine rotor diameter and the velocity at hub height, ranging from Re = 1.66 × 10⁴ to 1.73 × 10⁵. Results suggest that main flow statistics (mean velocity, turbulence intensity, kinematic shear stress and velocity skewness) become independent of Reynolds number starting from Re ≈ 9.3 × 10⁴. In general, stronger Reynolds number dependence was observed in the near wake region where the flow is strongly affected by the aerodynamics of the wind turbine blades. In contrast, in the far wake region, where the boundary layer flow starts to modulate the dynamics of the wake, main statistics showed weak Reynolds dependence. These results will allow us to extrapolate wind tunnel and computational fluid dynamic simulations, which often are conducted at lower Reynolds numbers, to full‐scale conditions. In particular, these findings motivates us to improve existing parameterizations for wind turbine wakes (e.g. velocity deficit, wake expansion, turbulence intensity) under neutral conditions and the predictive capabilities of atmospheric large eddy simulation models. Copyright © 2011 John Wiley & Sons, Ltd.     

Verification of computational simulations of the NREL 5 MW rotor with a focus on inboard flow separation

The aerodynamic characteristics of the NREL 5 MW rotor have been examined using a Reynolds‐averaged Navier–Stokes method, OVERFLOW2. A comprehensive off‐body grid independence study has been performed. A strong dependence on the size of the near‐body wake grid has been found. Rapid diffusion of the wake appears to generate an overprediction of power and thrust. A large, continuous near‐wake grid at a minimum of two rotor diameters downstream of the rotor appears to be necessary for accurate predictions of near‐body forces. The NREL 5 MW rotor demonstrates significant inboard flow separation up to 30% of span. This separation appears to be highly three dimensional, with a significant amount of radial flow increasing the size of the separated region outboard. A simple, continuous full‐chord fence was applied at the maximum chord location of the blade, within the region of separation. This non‐optimized device reduced the boundary‐layer cross‐flow and resulting separation. The fence increased energy capture by nearly 1% at a wind speed of 8 m s^?1 and slightly increased blade loading over the length of the span. Copyright © 2011 John Wiley & Sons, Ltd.