Edge Of Wake Research Articles

Large-eddy simulation of upwind-hill effects on wind-turbine wakes and power performance

The present study utilizes high-fidelity large-eddy simulation (LES) to examine the upwind-hill impacts on wind-turbine wakes and power performance. Both the LES and Reynolds-Averaged Navier–Stokes (RANS) show good agreement with the experimental results of average velocity, whereas the LES exhibits great improvements in predicting turbulence characteristics. In turbine simulations, the wake half-width shows different behaviors in the hill and flat cases. The Gaussian and cosine distributions are good representations of the unskewed profiles of velocity deficit, although some discrepancies are seen around the wake edges. However, the spanwise profiles of velocity deficit become skewed in some hill cases. The non-zero pressure gradient (NZPG) model is employed to predict the velocity deficit in the wakes of a hilltop-sited turbine, and a generalized condition is proposed so that the NZPG model can work in the scenario where the pressure gradients at the turbine location or in its near wakes are not zero. Appropriately assigning initial values for the NZPG model is important for its successful use. The presence of the upwind hill has an adverse influence on the power output of the hilltop-sited turbine. The installation of wind turbines in the valley region near upwind hills is not recommended due to the limited wind-energy resources and intense turbulence levels.

Self-similarity Characteristics of Vertical Axis Wind Turbine Wakes

Wake generated by wind turbine can greatly influence the performance of downstream turbine. To better understand the wake self-similarity characteristics of vertical axis wind turbine (VAWT), the shear stress transport (SST) turbulence model with the addition of the γ-Reθ transition model is performed to model a two-blade VAWT at different operating conditions. The simulated blade surface pressure and torque are compared with existing experimental results for validation. Results show that, the simulated results after considering the transition model are more consistent with the experimental results. Analysis of the flow field shows that the average streamwise velocity of the wake in the horizontal plane under different tip speed ratios is asymmetry, but symmetric in the vertical plane. Further analysis indicates that, at different downstream positions, the non-dimensional streamwise velocity deficit in the vertical plane remains self-similarity and basically coincides with the Gaussian distribution curve exclude the wake edges. In addition, the larger the tip speed ratio, the easier streamwise velocity deficit reach self-similarity state at downstream of the VAWT. The results of this study will be helpful to establish the wake model of the VAWT.

Development of an automatic thresholding method for wake meandering studies and its application to the data set from scanning wind lidar

Abstract. Wake meandering studies require knowledge of the instantaneous wake evolution. Scanning lidar data are used to identify the wind flow behind offshore wind turbines but do not immediately reveal the wake edges and centerline. The precise wake identification helps to build models predicting wake behavior. The conventional Gaussian fit methods are reliable in the near-wake area but lose precision with distance from the rotor and require good data resolution for an accurate fit. The thresholding methods, i.e., selection of a threshold that splits the data into background flow and wake, usually imply a fixed value or manual estimation, which hinders the wake identification on a large data set. We propose an automatic thresholding method for the wake shape and centerline detection, which is less dependent on the data resolution and quality and can also be applied to the image data. We show that the method performs reasonably well on large-eddy simulation data and apply it to the data set containing lidar measurements of the two wakes. Along with the wake identification, we use image processing statistics, such as entropy analysis, to filter and classify lidar scans. The automatic thresholding method and the subsequent centerline search algorithm are developed to reduce dependency on the supplementary data such as free-flow wind speed and direction. We focus on the technical aspect of the method and show that the wake shape and centerline found from the thresholded data are in a good agreement with the manually detected centerline and the Gaussian fit method. We also briefly discuss a potential application of the method to separate the near and far wakes and to estimate the wake direction.

Rethinking Automatic Ship Wake Detection: State-of-the-Art CNN-Based Wake Detection via Optical Images

Most existing wake detection algorithms use Radon transform (RT) due to the long streak features of ship wakes in synthetic aperture radar images. The high false alarm rate of RT requires that the algorithms have significant human intervention for image preprocessing. When processing optical images, these algorithms are greatly challenged because it is not only sea clutter that interferes with the wake detection in the optical waveband but also many other environmental factors. Therefore, in this article, we address the automatic ship wake detection task in optical images from the idea of the convolutional neural network (CNN)-based object detection and design an end-to-end detector named WakeNet as a novel method. WakeNet processes all the wake textures clamped by the V-shaped Kelvin arms as the object and conducts detection via the oriented bounding box; the additional regression of wake tip coordinates and Kelvin arm direction can improve the hard wake detection performance while predicting the wake heading. According to the X-shaped wake spectral pattern and the wake edge enhancement principle in adjacent scales, spectral attention and multiscale attention modules are also used. Furthermore, more than 11.6k wake optical images are collected and annotated, providing a benchmark dataset. The experimental results on the dataset show that the proposed method has a better performance than other CNN-based methods, and the detection results of the yacht wake polarization images further verify its practical value. This article demonstrates that the deep learning-based method has considerable advantages over the traditional RT-based methods in wake detection.

A vortex sheet based analytical model of the curled wake behind yawed wind turbines

Motivated by the need for compact descriptions of the evolution of non-classical wakes behind yawed wind turbines, we develop an analytical model to predict the shape of curled wakes. Interest in such modelling arises due to the potential of wake steering as a strategy for mitigating power reduction and unsteady loading of downstream turbines in wind farms. We first estimate the distribution of the shed vorticity at the wake edge due to both yaw offset and rotating blades. By considering the wake edge as an ideally thin vortex sheet, we describe its evolution in time moving with the flow. Vortex sheet equations are solved using a power series expansion method, and an approximate solution for the wake shape is obtained. The vortex sheet time evolution is then mapped into a spatial evolution by using a convection velocity. Apart from the wake shape, the lateral deflection of the wake including ground effects is modelled. Our results show that there exists a universal solution for the shape of curled wakes if suitable dimensionless variables are employed. For the case of turbulent boundary layer inflow, the decay of vortex sheet circulation due to turbulent diffusion is included. Finally, we modify the Gaussian wake model by incorporating the predicted shape and deflection of the curled wake, so that we can calculate the wake profiles behind yawed turbines. Model predictions are validated against large-eddy simulations and laboratory experiments for turbines with various operating conditions.

LES Study of Wake Meandering in Different Atmospheric Stabilities and Its Effects on Wind Turbine Aerodynamics

Wake meandering disturbs the stability of the far wake field and thus increases the fatigue loads of downstream wind turbines. A deep understanding of this phenomenon under atmospheric boundary layers and its relation to the structural loads helps to better model the dynamic wake and alleviate adverse effects. A large eddy simulation and an actuator line model are introduced in the present work to simulate the wake field and aerodynamic loads of wind turbines with different longitudinal spacings. By temporal filtering and the gaussian fitting method, the wake center and edge are precisely defined, and the dynamic wake characteristics, including the wake width, oscillation amplitude, and frequency, are described based on the statistical data of the simulated flow field. Results reveal that the wake meandering is caused by both large-scale atmospheric structure and the unstable vortex shed from the rotor because two distinct meandering frequency ranges are detected. As the atmosphere instability increases, the former becomes the dominant inducing factor of the meandering movements. Further, the analysis of the correlation between the inflow characteristics and the wake deflection shows that the Taylor hypothesis remains valid within a distance of over a thousand meters under both neutral and convective boundary layers, proving the feasibility of using this hypothesis for wake evolution prediction. In addition, our study shows that the fluctuation of blade root moment and yaw moment is significantly intensified by the meandering wake, with their standard deviation is augmenting by over two times under both atmospheric conditions. The power spectrum illustrates that the component with rotor rotation frequency of the former is sensible to the wake effect, but for the latter, the power spectrum density of all frequencies is increased under the meandering wake. These indicate that the fatigue loads will be underestimated without considering the wake meandering effect. Moreover, the high correlation between the wake deflection and yaw moment implies that we can predict yaw moment based on the incoming flow information with high accuracy.

Hydrodynamics features of dispersed bubbles in the ventilated wake flow of a cylinder

Abstract An experimental study was conducted to investigate the 2D bubbly flow downstream of a cylinder. Sparsely distributed bubbles were produced using the ventilation method. The carrier flow was measured using the particle image velocimetry (PIV) technique. The shadow imaging technique was used to capture instantaneous bubbly flow images. An image-processing code was compiled to identify bubbles in acquired image, calculate the bubble equivalent diameter and the bubble velocity. The effects of Reynolds number and the flow rate of the injected air were considered. The result indicates that the carrier flow is featured by distinct flow structures and the wake region is suppressed as the upstream velocity increases. Regarding the bubbles trapped in the wake flow, the number of small bubbles increases with the upstream velocity. On the whole, the bubble velocity is slightly lower than that of the carrier flow. The consistency between small bubbles and the carrier flow is high in terms of velocity magnitude, which is justified near the wake edge. The difference between the bubble velocity and the carrier flow velocity is remarkable near the wake centerline. For certain Reynolds number, with the increase in the air flow rate, the bubble equivalent diameter increases and the bubble void fraction is elevated.

Flow and coherent structures around circular cylinders in shallow water

Eddy-resolving numerical simulations are conducted to investigate the dynamics of the large-scale coherent structures around a circular cylinder in an open channel under very shallow flow conditions where the bed friction significantly affects the wake structure. Results are reported for three test cases, for which the ratio between the cylinder diameter, D, and the channel depth, H, is D/H = 10, 25, and 50, respectively. Simulation results show that a horseshoe vortex system forms in all test cases and the dynamics of the necklace vortices is similar to that during the breakaway sub-regime observed for cases when a laminar horseshoe vortex forms around the base of the cylinder. Given the shallow conditions and turbulence in the incoming channel flow, the necklace vortices occupy a large fraction of the flow depth (they penetrate until the free surface in the shallower cases with D/H = 25 and 50). The oscillations of the necklace vortices become less regular with increasing polar angle magnitude and can induce strong amplification of the bed shear stress beneath their cores. Strong interactions are observed between the legs of the necklace vortices and the eddies shed in the separated shear layers in the cases with D/H = 25 and 50. In these two cases, a vortex-street type wake is formed and strong three-dimensional effects are observed in the near-wake flow. A secondary instability in the form of arrays of co-rotating parallel horizontal vortices develops. Once the roller vortices get away from the cylinder, the horizontal vortices in the array orient themselves along the streamwise direction. This instability is not present for moderately shallow conditions (e.g., D/H ≈ 1) nor for very shallow cases when the wake changes to an unsteady bubble type (e.g., D/H = 50). For cases when this secondary instability is present, the horizontal vortices extend vertically over a large fraction of the flow depth and play an important role in the vertical mixing of fluid situated at the wake edges (e.g., by transporting the near-bed, lower-velocity fluid toward the free surface and vice versa). The largest amplification of the bed shear stress in the near-wake region is observed beneath these horizontal vortices, which means that they would play an important role in promoting bed erosion behind the cylinder in the case of a loose bed. Simulation results suggest that these co-rotating vortices form as a result of the interactions between the legs of the main necklace vortices and the vortical eddies contained into the newly forming roller at the back of the cylinder. The paper also analyzes how D/H affects the separation angle on the cylinder, the size of the recirculation bubble, the bed friction velocity distributions, and turbulence statistics.

Modelling lidar volume-averaging and its significance to wind turbine wake measurements

Lidar velocity measurements need to be interpreted differently than conventional in-situ readings. A commonly ignored factor is “volume-averaging”, which refers to lidars not sampling in a single, distinct point but along its entire beam length. However, especially in regions with large velocity gradients, like the rotor wake, can it be detrimental. Hence, an efficient algorithm mimicking lidar flow sampling is presented, which considers both pulsed and continous-wave lidar weighting functions. The flow-field around a 2.3 MW turbine is simulated using Detached Eddy Simulation in combination with an actuator line to test the algorithm and investigate the potential impact of volume-averaging. Even with very few points discretising the lidar beam is volume-averaging captured accurately. The difference in a lidar compared to a point measurement is greatest at the wake edges and increases from 30% one rotor diameter (D) downstream of the rotor to 60% at 3D.

Contributions of the Stochastic Shape Wake Model to Predictions of Aerodynamic Loads and Power under Single Wake Conditions

The contribution of wake meandering and shape asymmetry to load and power estimates is quantified by comparing aeroelastic simulations initialized with different inflow conditions: an axisymmetric base wake, an unsteady stochastic shape wake, and a large-eddy simulation with rotating actuator-line turbine representation. Time series of blade-root and tower base bending moments are analyzed. We find that meandering has a large contribution to the fluctuation of the loads. Moreover, considering the wake edge intermittence via the stochastic shape model improves the simulation of load and power fluctuations and of the fatigue damage equivalent loads. These results indicate that the stochastic shape wake simulator is a valuable addition to simplified wake models when seeking to obtain higher-fidelity computationally inexpensive predictions of loads and power.

Investigation into the wake aerodynamics of a five-straight-bladed vertical axis wind turbine by wind tunnel tests

Wake characteristics have significant effects on the performance design of standalone turbines and the optimal placement of multiple turbines. In the literature to date, little experimentation has been done on the wake of vertical axis wind turbines (VAWTs), and understanding of such wake is far from adequate. In this work, systematic measurements are presented of both the near and mid-range wake of a five-straight-bladed VAWT in a wind tunnel. The blockage ratio of the VAWT was 1.8%, and no correction of the measured data was required. The wake flow fields were measured up to 10 turbine diameters (10D) to the downstream. The wake exhibited high asymmetry in the horizontal direction. In addition, the wake expanded more in the horizontal direction than in the vertical direction. The causes of the asymmetry were analyzed and discussed through the experimental results. An engineering wake model was proposed to characterize the wake edges and the average velocities. The existence of a pair of counter-rotating vortical structures in the wake was detected. Moreover, the integral length scale was found to steadily grow with the downstream distance. This work contributes to the knowledge of the VAWTs׳ wake and the application of VAWTs in wind farm layout design.

A Large-Eddy Simulation Study of Vertical Axis Wind Turbine Wakes in the Atmospheric Boundary Layer

In a future sustainable energy vision, in which diversified conversion of renewable energies is essential, vertical axis wind turbines (VAWTs) exhibit some potential as a reliable means of wind energy extraction alongside conventional horizontal axis wind turbines (HAWTs). Nevertheless, there is currently a relative shortage of scientific, academic and technical investigations of VAWTs as compared to HAWTs. Having this in mind, in this work, we aim to, for the first time, study the wake of a single VAWT placed in the atmospheric boundary layer using large-eddy simulation (LES). To do this, we use a previously-validated LES framework in which an actuator line model (ALM) is incorporated. First, for a typical three- and straight-bladed 1-MW VAWT design, the variation of the power coefficient with both the chord length of the blades and the tip-speed ratio is analyzed by performing 117 simulations using LES-ALM. The optimum combination of solidity (defined as N c / R , where N is the number of blades, c is the chord length and R is the rotor radius) and tip-speed ratio is found to be 0.18 and 4.5, respectively. Subsequently, the wake of a VAWT with these optimum specifications is thoroughly examined by showing different relevant mean and turbulence wake flow statistics. It is found that for this case, the maximum velocity deficit at the equator height of the turbine occurs 2.7 rotor diameters downstream of the center of the turbine, and only after that point, the wake starts to recover. Moreover, it is observed that the maximum turbulence intensity (TI) at the equator height of the turbine occurs at a distance of about 3.8 rotor diameters downstream of the turbine. As we move towards the upper and lower edges of the turbine, the maximum TI (at a certain height) increases, and its location moves relatively closer to the turbine. Furthermore, whereas both TI and turbulent momentum flux fields show clear vertical asymmetries (with larger magnitudes at the upper wake edge compared to the ones at the lower edge), only slight lateral asymmetries were observed at the optimum tip-speed ratio for which the simulations were performed.

The turbulent wake of a towed grid in a stratified fluid

In a stable background density gradient, initially turbulent flows eventually evolve into a state dominated by low-Froude-number dynamics and frequently also contain persistent pattern information. Much empirical evidence has been gathered on these latter stages, but less on how they first got that way, and how information on the turbulence generator may potentially be encoded into the flow in a robust and long-lasting fashion. Here an experiment is described that examines the initial stages of evolution in the vertical plane of a turbulent grid-generated wake in a stratified ambient. Refractive-index-matched fluids allow optically based measurement of early ($Nt<2$) stages of the flow, even when there are strong variations in the local density gradient field. Suitably averaged flow measures show the interplay between internal wave motions and Kelvin–Helmholtz-generated vortical modes. The vertical shear is dominant at the wake edge, and the decay of horizontal vorticity is observed to be independent of $\mathit{Fr}$. Stratified turbulence, originating from Kelvin–Helmholtz instabilities, develops up to non-dimensional time $Nt\approx 10$, and the scale separation between Ozmidov and Kolmogorov scales is independent of $\mathit{Fr}$ at higher $Nt$. The detailed measurements in the near wake, with independent variation of both Reynolds and Froude numbers, while limited to one particular case, are sufficient to show that the initial turbulence in a stratified fluid is neither three-dimensional nor universal. The search for appropriately generalizable initial conditions may be more involved than hoped for.

Vortical structures and turbulent bursts behind magnetic obstacles in transitional flow regimes

The present paper reports on numerical investigations of vortical structures in transient flow regimes generated by the local action of the Lorentz force on an electrically conductive fluid. The locally imposed non-uniform magnetic field generates similar effects as observed for flows over submerged solid obstacles. It is demonstrated that complex flow patterns can be generated by imposing magnetic fields of different strengths. The initial validation of the electromagnetically extended Navier–Stokes solver on unstructured numerical grids is performed in the low-Reynolds number range 100 ⩽ Re ⩽ 400 for different values of the magnetic interaction parameter. A generally good agreement is obtained in comparison with similar numerical studies of Votyakov et al. (2007, 2008) for the low-Reynolds number cases. Then, a series of simulations are performed in transitional flow regimes ( Re = 900) for different values of the interaction parameter ( N = 3, … , 25). Simulations demonstrated the appearance of vortex-shedding phenomena similar to the flows behind solid obstacles. In contrast to the solid obstacles, the magnetic obstacles also generated the vortical flow patterns inside the magnetically affected regions. This feature can be used for the flow control of electrically conductive fluids, for efficient enhancements of the wall-heat transfer or for better mixing of passive scalars. Despite the laminar inflow conditions, turbulent bursts are observed in the magnetic wake region for the Re = 900 case. The velocity spectra and spatial distributions of the long-time averaged second-moments of the velocity field demonstrated that turbulence was locally sustained in the proximity of the magnetic wake edge.

The growth of a cylinder wake in turbulent flow

In the absence of ambient turbulence, the width of the downstream wake of a cylinder grows diffusively either through viscous effects or entrainment at the outer wake edge, depending on the characteristic Reynolds number, and in both the cases it increases as x 1/2, where x is the distance downstream of the cylinder. We examine the effect of ambient turbulence on the downstream wake of a cylinder, where this turbulence is treated as homogeneous and isotropic. Wake spreading becomes affected when the velocity deficit has decayed sufficiently that it is comparable with the root mean square velocity of the ambient turbulence. A new model is developed to explain how and when the ambient turbulence is sufficiently strong, and the wake is passively advected by the turbulence in the freestream. In this case, when the wake width is smaller than the integral length scale, the wake grows linearly with distance downstream (i.e. proportional to x), whereas when the wake width is larger than the integral length scale, it grows diffusively (i.e. proportional to x 1/2). A laboratory study of the decay of the velocity deficit behind a cylinder confirms the salient aspects of the new model predictions.

基于MATLAB的尾流图像数字化处理

The working principles and methods of wake images digital processing system were introduced.The gray-scale transformation,filtering,Canny marginalization,binary transformation and histogram transformation of two wake flow images were carried on,the speed of which are 30 km/h and 40 km/h respectively,using MATLAB software,experimental results were analyzed,and the speed of the ship wake edge angle,edge curvature and histogram peak change were obtained.The research results provide a basis for wave images entering the informationization system.

A Wind-Tunnel Investigation of Wind-Turbine Wakes: Boundary-Layer Turbulence Effects

Wind-tunnel experiments were performed to study turbulence in the wake of a model wind turbine placed in a boundary layer developed over rough and smooth surfaces. Hot-wire anemometry was used to characterize the cross-sectional distribution of mean velocity, turbulence intensity and kinematic shear stress at different locations downwind of the turbine for both surface roughness cases. Special emphasis was placed on the spatial distribution of the velocity deficit and the turbulence intensity, which are important factors affecting turbine power generation and fatigue loads in wind energy parks. Non-axisymmetric behaviour of the wake is observed over both roughness types in response to the non-uniform incoming boundary-layer flow and the effect of the surface. Nonetheless, the velocity deficit with respect to the incoming velocity profile is nearly axisymmetric, except near the ground in the far wake where the wake interacts with the surface. It is found that the wind turbine induces a large enhancement of turbulence levels (positive added turbulence intensity) in the upper part of the wake. This is due to the effect of relatively large velocity fluctuations associated with helicoidal tip vortices near the wake edge, where the mean shear is strong. In the lower part of the wake, the mean shear and turbulence intensity are reduced with respect to the incoming flow. The non-axisymmetry of the turbulence intensity distribution of the wake is found to be stronger over the rough surface, where the incoming flow is less uniform at the turbine level. In the far wake the added turbulent intensity, its positive and negative contributions and its local maximum decay as a power law of downwind distance (with an exponent ranging from −0.3 to −0.5 for the rough surface, and with a wider variation for the smooth surface). Nevertheless, the effect of the turbine on the velocity defect and added turbulence intensity is not negligible even in the very far wake, at a distance of fifteen times the rotor diameter.

The application of wavelets correlator for ship wake detection in sar images

The detection of the wake can provide substantial information about a ship, such as its size, direction, and speed of movement. In general though, ship-generated wakes in synthetic aperture radar images are associated with high sea clutter, which will cause some deterioration in the detection performance. Therefore, a wavelet correlator, based on an orthogonal basis function, is adopted. Three highpass images - horizontal, vertical, and diagonal direction - are generated for each resolution scale, followed by a process to correlate among the moduli of different scale modulus images formed from the three highpass images. The output of the correlation process is highly representative at the ship's wake edges. Comparisons with other methods indicate the superior performance of the present approach, in that not only can the wakes be detected, but their V-shaped pattern is well preserved. The second stage involves the application of the Radon transform technique to an estimation of the V-opening angle from the detected ship wakes. Ship-generated wake edges are found to be the local maxima in the wavelet transform method of several adjacent scales, and hence, the wake edge will be enhanced in the reconstructed data. The background noise is also greatly reduced. In particular, the process of spatial correlation is found to be critical. Compared to a direct Radon transform, the proposed scheme is demonstrated to be much more effective in terms of efficiency, as well as reliability, for ship wake detection in noisy backgrounds.

Ship wake detection in synthetic aperture radar images using a combination of a wavelet correlator and Radon transform

A moving ship in a synthetic aperture radar (SAR) image produces a multiscale wake with a characteristic linear V-shaped pattern. Detection of the wake can provide substantial information about the ship, such as its size,direction, and speed of movement. In general, ship-generated wakes in SAR images are associated with high sea clutter, which causes the deterioration of detection performance. We present a hybrid method that combines the wavelet technique and the Radon transform technique to detect the ship wake. The wavelet technique is first applied to generate a set of multiscale images. An orthogonal basis function is adopted so that three high-pass images in the horizontal, vertical, and diagonal directions are generated for each resolution scale. Then a spatial correlator is applied among the moduli of different scale, where modulus images are formed from three high-pass images. The output of the correlation process is shown to be highly representative at ship wake edges. Comparisons with other methods indicate the superior performance of this approach in that not only are the wakes detected, but the V-shaped pattern is well preserved. The second stage of the method involves the application of the Radon transform technique to estimate the V opening angle from the detected ship wakes. Compared with a direct Radon transform, the greater effectiveness of the proposed scheme is demonstrated in terms of efficiency as well as reliability for ship wake detection in noisy backgrounds.

Unsteady 3D nonlinear Kutta-Joukovski condition for thin lifting surfaces

Considering unsteady potential flows, the fluid velocity regularity condition at a wake shedding edge – often referred to as the Kutta-Joukovski condition – has been clarified in the 2D nonlinear and 3D linearized cases only, and the usual nonlinear direct numerical approaches, as they are built on an already discretised view of the problem, skip over the theoretical question. A nonlinear Kutta-Joukovski condition is proposed here for unsteady 3D flows around thin airfoils, for which the body and its trailing-edge-shed wake are represented by double layers. The fluid velocity field, deriving from the potential function, is expressed using the two usual terms, relating to the edge of the sheet and its surface. Removing the singularity for the first term leads to the well-known condition for the potential jump across the shedding edge to be continuous from the body to the wake. For the second, works by Legras, for the steady case – and readily extendible to the unsteady one – exhibit its logarithmic singular behavior and allow to derive a regularity condition that deals with the surface gradient of the potential jump across the shedding edge. Lastly, implemented in a general non linearized situation, these conditions lead to an expression relating the geometric and doublets distribution characteristics of the two surfaces (wing and wake).