The contribution of large scale structures in the power generation of finite scale wind farms using large eddy simulation

Abstract

The large scale organizations in the flow around the finite scale wind farms that contribute to the turbine power, have been studied in the current paper. The study has been carried out using Large Eddy Simulation (LES) with near wall modelling, and the turbine forces are modelled using the actuator line model. Proper orthogonal decomposition (POD) has been used as a tool of analysis to understand the large scale features contributing to the power generation by wind turbines in different rows of a wind farm. The POD modes reveal the existence of energetic flow features significantly larger than the turbine rotor diameter contributing to the flux of the mean kinetic energy (MKE). Thes fluxes play an instrumental role in power generation as also observed in the previous literature. New insights on the flow structures around the wind farm have been obtained which opens up further research directions to understand the localized transfer of the MKE flux. INTRODUCTION Large wind farms in atmospheric boundary layer (ABL) are often studied in the asymptotic limit of infinite number of wind turbine rows, where the flow past the wind turbines is fully developed given by periodic boundary conditions in the streamwis and spanwise directions as seen in Frandsen et al. (2006), Calaf et al. (2010), and describable through simple equilibrium laws. This assumption essentially neglects many complex flow features like the growth of the inner layer due to the turbulent dispersion of the wakes, impingement of the wakes from one row of wind turbines to the next row and beyond, which typically results in a decreased power and an increased structural loading of the downstream turbines. More importantly, these flow features which arise due to the spatial variability of the convection of kinetic energy, cannot be neglected for wind farms where the streamwise and spanwise extent of the layout are comparable to the atmospheric boundary layer thickness. The previous literature has shown the contribution of large scale structures responsible for the power generation in infinite wind farms, e.g., using Fourier analysis by Chatterjee & Peet (2016b) and proper orthogonal decomposition (POD) by VerHulst & Meneveau (2014). Hamilton et al. (2016) looked at proper orthogonal decomposition of an experimental database, but they also focused on a homogeneous part of the wind farm beyond the fourth row, that can be approximated by the fully developed condition and row-to-row periodicity. To the authors knowledge, no such study has been performed to understand the behaviour of large scale features in the power generation of aperiodic, finite scale wind farms. Understanding the multi scale dynamics involved in the interaction of large scale atmospheric flows with the wind turbine rotors in the first and subsequent rows is important, as this will improve our interpretation of power generation in wind turbine arrays required for an efficient optimization of the wind farm layout. For infinite wind farms with homogeneity in the horizontal direction, Fourier analysis is a natural choice for studying the length scales of motion. However, in finite scale wind farms, the streamwise inhomogeneity due to the inner layer growth, renders the use of Fourier analysis to be limited. Consequently, in the current paper, we propose to present the POD analysis of the flow features past the wind turbine array, and also to understand the variation of large scale modal structures that contribute to the power compared to infinite wind farms. NUMERICAL METHOD The numerical method implements a variational formulation of the Navier-Stokes equations involving Galerkin projection using an exponentially accurate open-source spectral element (SEM) solver Nek5000 (See Fischer et al. (2008)). The numerical simulations at Reynolds number 1010 based on the hub-height velocity and the boundary layer thickness are carried out using Large Eddy Simulation (LES) with near wall modelling, and the wind turbine forces are modelled using the state-of-the-art actuator line model (Refer to Chatterjee & Peet (2016a,b) for details of the model). To provide a validation of our spectral element LES near-wall modeling methodology, we plot the non-dimensional streamwise velocity gradient and streamwise kinetic energy spectra for the LES simulations of a neutral ABL in Figure 1 that shows correct logarithmic trends of the streamwise velocity profile as well as the appropraite scaling laws of k−1 x (overlap between inertial and integral scales) and k x (overlap between inertial and dissipation scales) as measured by Perry et al. (1986). The computational domain is rectangular and cartesian, of the size 3πH×πH×H in streamwise, spanwise and wall-normal directions, respectively, with H being the ABL thickness. The domain consists of a 3× 3 organized array of wind turbines, with the rotor diameter D = 0.2H. The streamwise and spanwise inter-turbine distances are 7D and 3D respectively, with the first row of turbines placed πH distance from the inlet boundary. The spanwise boundary conditions are periodic, the top boundary condition is stress free and the bottom boundary has a shear-stress boundary condition corresponding to the near wall modelling of the log law of the wall (Chatterjee & Peet (2016a,b)). The inflow boundary condition is generated from a separate precursor simulation of a neutral periodic ABL flow with the domain size 2πH × πH ×H, while stabilized outflow boundary conditions have been used at the streamwise outlet of the wind turbine domain (Refer to Chatterjee & Peet (2016a) for details of inflow-outflow boundary condition). LES simulations of a flow past the 3× 3 wind turbine array with the inflow-outflow boundary conditions were performed for the duration of 100 flowthrough times to ensure statistical stationarity, after which snapshots and statistics were collected for 100 more flow through times for the POD analysis.