Near wake region of an industrial scale wind turbine: comparing LES-ALM with LES-SMI simulations using data mining (POD)

Accurate prediction of power generation capability needs proper assessment of blade loading and wake behavior. In this regard, the Sliding Mesh Interface (SMI) approach and the Actuator Line Model (ALM) are two diverse computational fluid dynamics (CFD) based approaches of simulating the turbine behavior, each having its own merits and demerits. The SMI technique simulates the unsteady flow by explicitly modeling the blades and their rotation using a dynamic mesh, while in Actuator Line Model, the blades are not modeled explicitly but each blade is resolved as a rotating line (made of N actuator segments), over which the forces are computed. The current work focuses on simulating an industrial scale reference turbine and in differentiating the near wake dynamics predicted by these two approaches using Large Eddy Simulation (LES) and Proper Orthogonal Decomposition (POD) technique (a data mining tool). Initially, the ALM is compared with FAST model for the prediction of variation of power coefficient with the Tip Speed Ratio (TSR). The ALM is able to capture the varying trend and it predicts a similar optimum tip speed ratio as the FAST model. At this optimum TSR condition, the ALM is compared with the SMI method for a study limited to the near wake region. Comparisons between SMI and ALM shows that : (a) The SMI is predicting more complex 3D nature of the flow, and (b) the POD shows that ALM captures the shear regions of wake but it does not capture the vast compendium of length and time scales of eddies as SMI does. However, despite these limitations, the ALM has been able to capture the qualitative trend in wake deficit and the power coefficient variation with tip speed.


Introduction and objective
Wake dynamics have been shown to influence the power production capabilities of downstream turbines depending upon the inter-turbine distance in a wind farm layout [1][2][3][4][5]. Actuator Line Model (ALM) with Large Eddy Simulation (LES) turbulence model has been used popularly to understand the wake dynamics in wind farms as it is computationally tractable to do so for multi-turbine set-ups. However, it is well-known that ALM does not explicitly resolve turbine blades and hence might not be expected to be accurate enough. On the other hand, the Sliding Mesh Interface (SMI) approach resolves the blade and is expected to be more accurate but it is computationally intractable to perform for a set-up with multiple turbines. In this regard, the current work compares the SMI and ALM methods for one industrial scale turbine. The NREL 5MW reference turbine [6] is chosen for this work, as it is a realistic and standardized industrial-scale off-shore turbine model. NREL 5 MW turbine [6] [7] consists of three 63m long blades defined in terms of eight cross sectional profiles (DU21, DU25, DU30, DU35, DU40 and NACA64) and twist angles at different locations away from the hub (as described in [6]). These multiple-sections with diverse angles of attack provide an ideal opportunity to test and benchmark models and methodologies that can later be applied to solve a bigger range of industrial challenges. Hence, it is popularly used by leading groups (several of USA DOEs Wind and Hydro-power Technologies Programs, EU's UpWind research program, and the International Energy Agency (IEA)'s Wind Annex XXIII Subtask Offshore Code Comparison Collaboration) to test methodologies. Such industrial scale wind-turbine blades comprising of multiple sections exhibit complex flow-patterns along the blade length [8] and generate wakes influencing wind-farm operations. In this regard, a comparison of wake dynamics and power production simulation results between sliding mesh approach (SMI) and Actuator line model will be useful to understand the comparative predictive performance of model. Thus, the objectives of this work are : 1.1. Objectives (i) Compare the performance of ALM-LES turbulence model with FAST model in predicting power coefficient variation with varying tip speed ratio. (ii) Compare the performance of ALM-LES turbulence model with SMI-LES approach in capturing wake dynamics at the optimum tip speed ratio.

Approach and Methods
The study is limited to the near wake region. The approach involves comparison of ALM and SMI approaches. Initially, the ALM is compared with FAST model for the prediction of variation of power coefficient with Tip Speed Ratio (TSR). This is done for the NREL 5 MW reference turbine. Then at the optimum TSR, ALM and SMI are compared for predicting wake structure and flow patterns. The wake dynamics simulated by SMI and ALM for the optimim T SR = 7.5 are compared by analyzing mean wake deficit profiles and by using Proper Orthogonal Decomposition (POD). Due to lack of experimental data of wake deficit velocity profiles for NREL 5 MW reference turbine, it is not possible to do validation of the models, so we resort to a verification exercise (i.e. comparison of different models). The governing equations and application details of SMI, ALM and POD is explained next section.

Sliding Mesh Interface (SMI) approach
The conservation equations are solved on a moving mesh (a sliding mesh) to account for the impeller motion and to capture unsteady interactions between rotating part and stationary bafes. This approach involves dividing the computational domain into a rotating and a stationary zone using a sliding interface. The approach assumes that the rotating computational mesh moves (or slides) relative to the stationary frame. The terms of the governing equations are integrated on a control volume, and the effect of the moving control volume is accounted by including the mesh motion flux in the computation of face mass flux in the convective terms (in other words the convective terms in the governing equation are modied). Thus, the face mass fluxes in the convective terms have to be computed relative to the mesh motion flux (ie. the volume swept by moving cell face during its movement with the cell face velocity). This mesh motion flux is computed based on the space (or volume) conservation law, which ensures that the moving face velocity is calculated from the face-centered positions, such that the surface vectors as well as calculation volumes inside rotating part remain constant. The mesh position in the rotating domain is updated after every time step (as it changes with impeller rotation) and so are the cell face positions at the sliding interface. At the sliding interface, a conservative interpolation is used for both mass and momentum using a set of fictitious control volumes. The governing equations are solved only in an inertial reference frame. Following equations are solved throughout the domain for SMI approach using LES turbulence model.

Actuator Line Model (ALM)
The turbine is modeled using actuator line model (ALM) approach, which was first developed by Sørensen and Shen [9]. The actuator line model ( accounts for the fact that the force exerted by turbine on the flow field is equal and opposite to the force experienced by it due to the flow. At the location (x, y, z) of the fluid domain, the overall body force is summation of force over all N actuator segments of the turbine, where (x j , y j , z j ) is the location of the j th segment and r j is the distance between segment j and the fluid domain location.
Numerical implementation -Computational domain, mesh and parameters The turbulence is modeled using a one-equation sub-grid scale (SGS) turbulent kinetic energy LES model for both SMI and ALM. Figure 1 shows the computational domain, mesh and suitability of mesh. The adequacy of mesh for LES Simulations can be judged through the contour of the ratio of sub-grid-scale-kinetic-energy k sgs to total-kinetic-energy k total (as seen in figure 1(c)-1(d)). In LES simulations, the mesh-size determines spatial filtering and establishes cut-off between resolved and unresolved (modeled sub-grid scale) parts of flow. Finer the mesh, more part of flow is resolved and more accurate are the simulation. As per criteria of Pope (2000) [10], for a well-resolved LES, less than 20% of the total kinetic energy should be modeled subgrid-scale part (i.e. k sgs /k total ratio should be less than 0.2). Figure  . This grid should be sufficient to resolve wake tip vortex [11][12][13]. For SMI, the blade surface has been treated as wall with no-slip boundary and employs a wall function based on Spalding's law [14] that gives a continuous kinematic viscosity profile to the wall over wide range of y + . This is required because the average y + value near blade wall is 667 while the minimum y + value is 11 and in terms of grid size -the minimum wall normal grid size is 0.008m near blade. Simulations are conducted with a uniform inlet wind velocity of 9m/s applied on the inlet face at different T SRs. The T SR is changed by adjusting the rotational speed of the turbine while keeping the inlet wind velocity constant. At the outlet face a standard outlet boundary condition is used (fixed pressure value and zero normal gradient for rest). A free slip boundary condition is applied on the rest of the surfaces. For the actuator line method, about 40 actuator segments (i.e parameter N in equation 5) were used in the simulation. The regularization parameter (ε in equation 5)in ALM is chosen to be about two times the cube root of grid volume size in that region. This is selected so as to ensure that the force is not overly concentrated to cause numerical oscillations / solver instability, and neither does the force becomes too smoothed so as to cause no resistance to the wind flowing through the turbine.  In this work, all the equations (except k sgs ) use second order linear convection discretization scheme. Similarly, the gradient term computation at cell faces accounts for both the orthogonal and the non-orthogonal parts.

Proper Orthogonal Decomposition (POD)
For the computation of the POD modes two dimensional snapshots of any variable (velocity components here) is required. The N snapshots are represented by U = [u 1 , u 2 · · · u N ] which is used to compute the covariance matrix given by C = U T U. After this an eigenvalue problem CA i = λ i A i is solved to obtain the eigenvalues λ i and eigen vectors A i which are sorted in a decreasing order as λ 1 > λ 2 > · · · > λ N . POD modes are then computed as With POD modes arranged as Ψ = [φ 1 φ 2 · · · φ N ]. POD coefficients a i can be found from the snapshot n as a n = Ψ T u n . From this a snapshot can be reconstructed as u n = Ψa n . Relative energy given by any i th mode is given by λ i / N j=1 λ j . POD has been successfully applied to understand wake dynamics [15,16].

Verification study: Power coefficient (C p ) Vs tip speed ratio(TSR)
For the NREL 5 MW turbine, Jonkman et.al. ([6]) studied the evolution of the power coefficient as a function of the T SR and blade-pitch surface by running FAST [17] with AeroDyn simulations. From these simulations, they found that the peak power coefficient of 0.482 occurred at a T SR of 7.55 and a rotor-collective blade-pitch angle of 0.0 • . Similar study was carried out by Rannam Chaaban (data available at https://wind.nrel.gov/forum/wind/viewtopic.php?t=582) and he predicted the optimum C p to occur at T SR = 8. Furthermore. Rannam's study has been used for the verification of our ALM and SMI model in Figure 2. The ALM method follows a similar trend and gives optimum power coefficient at a the T SR of 7.5 but it over-predicts the optimum power coefficient value (around 0.56). Figure 2 shows a comparison between the variation in power coefficient predicted by ALM and FAST methods. It is possible to obtain similar results as FAST by tuning regularization parameter . However, this is not done in the current work as it's a verification exercise and not a validation exercise, and all models can differ from the exact experimental observation (which are currently not available for NREL 5 MW reference turbine). The SMI method predicts a C p = 0.54 at T SR = 7.5. The next verification is done for wind deficit.

Wake dynamics with tip speed ratios
The ALM has also been used to compare the impact of T SR on mean wind deficit (figure 4). an optimum T SR of 7.5, more energy is extracted by the blades in that region leading to higher wind deficit and also higher C p , while as we increase or decrease the tip speed ratio (towards 9 or 6), the energy extracted is lesser resulting in lower wind deficit and lower C p . The wake deficit by SMI for TSR 7.5 is compared with ALM in (figure 4), and it differs in two ways, firstly SMI predicts a higher wind deficit in the near hub region (0.1 > z/R) at locations 0.3R, 0.9R and 1.3R downstream of turbine. This is because the hub is explicitly modeled in case of SMI, whereas in ALM, the hub region experiences higher wind velocity and a lower negative wind deficit (0.1 > z/R), secondly in regions near tip vortex (0.9R > z/R > 1.1), ALM shows a sharp velocity gradient corresponding to the presence of shear region accompanying tip vortex and it shows higher wake velocity deficit in the core wake region ( 0.2R > z/R > 0.9). Either, it could be because of higher turbulence that the wakes are decaying faster in SMI or because of mesh size. Despite these differences in wake velocity deficit, the power coefficient in SMI is slightly lower than ALM because higher energy gets extracted at the tip edge region (which produces most of the torque) which compensates for the lower energy extracted at the core wake region in SMI, as compared to the ALM case. Figure 3 compares the flow structures and flow pattern captured by SMI and ALM Approach. The wake structures obtained in the near wake region by SMI and ALM are shown in figure  3(a) and in figure 3(d) respectively. The vorticity contours show that both the methods (and the mesh used) are able to capture the helical wake structures downstream of blade at an altitude along the periphery of blade tip. A higher vorticity region is captured by SMI at regions downstream at altitude between the tip periphery and root, which could be because of a chaotic flow induced in this region by explicit rotating blade of SMI as shown in streamlines (figure 3(e)). Figure 3(b) and figure 3(e) compares the flow path (streamlines) predicted by SMI and ALM, with streamlines colored by magnitude of velocity. SMI captures a higher 3D chaotic flow behavior in regions behind the turbine as compared to the ALM (which does not show this much chaotic 3D flow). Further, the highest velocity in ALM streamline is about 11 m/s while SMI streamline captures a highest velocity (65 m/s) almost equivalent to the blade tip speed (ωr = 1.071 * 63 = 67.4m/s), which could be due to non-slip boundary on the rotating blade geometry in SMI. For sake of comparison, the color-bar range for streamlines is set at the highest velocity magnitude captured by the SMI. Further, figure 3(c) and figure 3(f) shows the flow pattern at the DU40 segment (located near the hub) in terms of mean velocity vectors captured by the SMI and ALM respectively. In figure 3(f), the velocity vectors colored by velocity magnitude is imposed on a pressure contour. At TSR 7.5, SMI shows the wind approaching the blade at an angle of attack and the explicit rotating bluff body airfoil segment has major influence on the velocity vector direction and magnitude as it crosses the blade, while with ALM, the velocity vectors seemed to only deviate slightly as it crossed the similar blade region (note that ALM do not have explicit blade and this DU40 region was identified using vorticity iso-surface). The flow vector differences can be attributed to the distributed nature of force projection in ALM methodology along with the non-explicit resolution of blade. However, despite these limitations, ALM has been able to predict the trends in wake deficit and power coefficients. Thus, the performance of ALM method is compared with the SMI model. The impact of these two diverse methodologies on wake dynamics is further captured by POD as described in section 3.3 below.

POD comparison
To conduct POD the simulation data from ALM and SMI was first interpolated onto a rectilinear grid. For ALM, the data sampled at 7 Hz, is interpolated on a uniform rectilinear grid measuring 380 × 400 elements, with a grid-size of 0.5m × 0.5m. The POD is applied on 720 planes over a total time of 100s. For SMI, the data sampled at 166 Hz, is interpolated on a uniform rectilinear grid measuring 240 × 400 elements, with a grid-size of 0.5m × 0.5m. The POD is applied on 1000 planes over a total time of 6s (corresponding to nearly one rotation). It is ensured that most of the flow realizations are captured in the samples used for conducting the POD. gives the flow structures represented by the first four modes for the ALM simulations conducted for at T SR = 7.5. It can be seen from the figure that the first mode represents the most energetic (97.5% of the total energy) and probable realization of the flow which looks like the mean wind field. As we move to higher modes we discover that the associated energy decrease rapidly. Together, the first three modes account for 99.5% of the total energy. The spatial scales associated with flow structures also decrease. It is also clear from the figures that the first four modes capture distinct kinds of flow structures hinting at a clear scale separation at least in space. On the other hand, in the case of SMI, the first mode corresponds to only 90% of the total energy. It hints at a spreading of the larger flow structures to higher modes. The explicit churning motion of the blades generate large eddies (Figure 7) which break down into smaller eddies and no clear scale separation is observed when compared to the ALM simulation as a result of which the energy drop across modes is more gradual. However, together the first 30 modes capture 99.5% of the total energy. The need for slightly more number of modes in the case of SMI suggests that it is able to capture flow with more unsteadiness and wider range of spatio-temporal scales either owing to explicit blade resolution or perhaps the faster decay of wakes through the energy cascading process (and hence there are more length and time scales in the given domain). It is worth highlighting here that although the effects of hub vortex is quantitatively captured using both the approach, the same is not true for the tip-vortices. A reasonable explanation can be that the current POD is conducted on 2D snapshots where the tip In order to capture the helical tip vortices, POD using three dimensional field should be conducted. This was not possible during the course of this work but remains an investigation for the future.   Figure 7. First four modes extracted from the SMI simulations using POD.Contours of Decomposed velocity modes.Red color region represents high values and blue color region represents low values.

Conclusion and future work
The paper aims to use LES and POD technique to differentiate wake dynamics captured by two diverse methods (ALM and SMI techniques) for an industrial scale wind turbine. The paper has been able to show that: • ALM method is compared with the FAST model in its ability to compute power coefficient with varying tip speed ratios for an industrial scale turbine, and ALM is able to capture the trend in a similar way as FAST. • The POD analysis also reveals that SMI has been able to capture a compendium of eddies with variable time scales. As a result, its first mode is not as dominant (90% of energy) as . The ALM first mode shows the shear along the tip periphery and hub periphery dominantly and most energy lies in this mode itself, while the rest of structures (scales) have less energy and are in the remaining modes. • A comparison in prediction of flow-pattern and wake structure by ALM and SMI shows that the SMI predicts a more complex 3D nature of flow. Despite these differences, the ALM has been able to follow the qualitative trend in wake deficit and power coefficient variation with tip speed, owing to the line source and the distributed nature of force projection. The current work is focussed on near wake region study. In future, it will be interesting to study the wakes further downstream.