A vortex particle-mesh (VPM) method with immersed lifting lines has been
developed and validated. Based on the vorticity–velocity formulation of the
Navier–Stokes equations, it combines the advantages of a particle method and
of a mesh-based approach. The immersed lifting lines handle the creation of
vorticity from the blade elements and its early development. Large-eddy simulation (LES) of vertical axis wind turbine (VAWT) flows is performed. The complex wake development is
captured in detail and over up to

The aerodynamics of vertical axis wind turbines (VAWTs) are inherently
unsteady, which leads to vorticity shedding mechanisms due to both the lift
distribution along the blade and its time evolution. This translates into a
wake topology that is far more complex and unsteady than for VAWTs' horizontal axis counterparts (HAWTs), a characteristic which could be indicative of more
intense wake decay mechanisms for VAWTs. Additionally, their inherent
insensitivity to wind direction changes suggests a more robust efficiency in
turbulent conditions. Naturally, both traits have led to several claims of an
advantage of VAWTs over HAWTs in wind
farms

The coarse-scale aerodynamics and the wake of the VAWT are simulated using a
massively parallel implementation of a vortex particle-mesh flow solver. The
present method relies on the large-eddy simulation in the vorticity–velocity
formulation for incompressible flows (

The right-hand sides of these equations are evaluated efficiently on an
underlying mesh

The SGS model is a simplified version of the variational multiscale (VM)
model

In order to carry out the computational steps above, information is made
available on the mesh, and recuperated from the mesh, by interpolating back
and forth between the particles and the grid using high-order interpolation
schemes. Fortunately, this hybridization does not affect the good
numerical accuracy (in terms of diffusion and dispersion errors) and the
stability properties of a particle method. The present method does indeed still
waive the typical Courant–Friedrichs–Lewy (CFL) constraint for the explicit time integration of
advection,

This last discussion actually pertains to the issue of Lagrangian distortion
in particle methods. If left alone, particles can be seen to deplete regions
of the flow or cluster in regions. Several remedies have been proposed.
Dissipative terms can be added to the particle ordinary differential equations (ODEs) in order to limit the
particle deformations

The generation of vorticity along the blades is accounted for through an
immersed lifting line approach

The standard lifting line and the actuator line techniques are not able to
capture flow curvature effects. Indeed, if the flow relative to the blade is
curved, as is the case here for a blade in rotation through essentially
straight streamlines, the airfoil behaves as an airfoil with an additional
camber

We first present validation results against recent
work

Validation: evolution of the normal,

We report on VPM simulations with and without a curvature correction, which
here amounts to an inward pitch

The remainder of this section focuses on a low-solidity H-VAWT studied
numerically by

Figure

H-type VAWT with AR

H-type VAWT with AR

Figure

The off-design operating points exhibit the expected behaviors: a high TSR
will lead to smaller angles of attack and a decreased torque production, while
the low TSR causes a distinctive stall in the upstream region and also in the
downstream one. It is visible in the sharp transitions of the force
coefficients at

H-VAWT global performance: effects of aspect ratio, TSR, and spatial resolution.

H-type VAWT with AR

The instantaneous wakes of the AR

H-type VAWT with AR

The VAWT wake decay is of course also governed by the TSR in a fashion very
similar to that of the HAWT: a high TSR (Fig.

Effect of aspect ratio at

The behaviors of the upstream tip vortices within the rotor are more complex
to apprehend, as they are affected by several factors: the intrinsic roll-up
dynamics of a vortex sheet (with a time-varying strength) and the velocities
induced by the surrounding vortical structures, including the bound vortices
on the blades. To some degree, the latter can be crudely linked to the
overall rotor loading (the

Blockage is but one factor, however, and it is a global one. The discussion can be
refined as additional, and less immediate, effects are to be expected from
the machine geometry. The aspect ratio, as indicated by

H-type VAWT with AR

The average behavior of these wakes is studied through the mean axial
velocity

Finally, the averaged wakes exhibit a slight deviation in this midplane. As
expected, the behaviors of the three TSRs do correlate with the signs and
values of the side forces produced by the rotor (see

H-type VAWT with AR

The mean streamwise vorticity at three transverse slices is shown in
Fig.

H-type VAWT with AR

H-type VAWT: dimensionless displacement and momentum surfaces as functions of the streamwise coordinate.

We apply classical turbulent wake diagnostics to the characterization of the
wake decay. More specifically, we adapt integral quantities, such as the
displacement and momentum widths, to the present context; the wakes considered do indeed lack symmetry and exhibit strong secondary flow structures,
which makes the definition of a velocity deficit evolution based upon a
single characteristic point unsuitable. Thus, we define dimensionless
displacement and momentum surfaces, respectively, as

The dimensionless momentum surface

Finally, the case

A vortex particle-mesh method, here briefly presented, has been applied to large-scale and high-resolution large-eddy simulation (LES) of VAWT wakes. The method is capable of tracking vortical structures over very long times and distances. This has led to several insights into the vortex dynamics at work inside the wakes of VAWTs. The mean flow topology has been extracted; unsteady flow aspects, three-dimensional effects and classical wake diagnostics have also been studied. The impact of several of these flow features for the deployment of VAWTs in wind farms is considerable: the aspect ratio and the operating conditions of the machine greatly affect the wake decay and even allow the presence of a recirculation region. The present study merely constitutes a preliminary study of VAWT wakes. Direct follow-up work will investigate the 3-D topology of the averaged wake and its unsteadiness. We will then also consider the behavior of these machines and of their wakes in a turbulent wind. Our methodology can also accommodate rotor dynamics models and realistic controllers; this will bring definitive answers to the smoothness of torque generation for H-type VAWTs and their performances in wind farms.

The immersed lifting line VPM code and its Fourier-based
solver library are proprietary. The Parallel Particle-Mesh (PPM) library is
an open-source library

The data sets involved in this study consist of massive 3-D and time-dependent data sets, the handling of which is not tractable on a data registry. Readers interested in the raw simulation data or the post-processed statistics are invited to contact the authors.

PC and MD prepared and ran the simulations, and DGC performed their post-processing. PC and MD developed the code; YM and DGC developed the dynamic stall model inside the code. PC, MD, and GW contributed to the analysis and the discussion of the results. PC prepared the paper with contributions from all co-authors.

The authors declare that they have no conflict of interest.

The authors acknowledge the fruitful discussions with Thierry Maeder, Stefan Kern, and Dominic von Terzi at the Aerodynamics and Acoustic Lab at GE Global Research, Garching bei München. Matthieu Duponcheel was partially supported by the ENGIE-funded research project Small Wind Turbines. The development work benefited from the computational resources provided by the supercomputing facilities of the Université catholique de Louvain (CISM/UCL) and the Consortium des Équipements de Calcul Intensif (CÉCI) en Fédération Wallonie Bruxelles (FWB) funded by the Fond de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under convention no. 2.5020.11. The production simulations used computational resources made available on the Tier-1 supercomputer of the FWB, infrastructure funded by the Walloon Region under grant agreement no. 1117545. Edited by: Carlo L. Bottasso Reviewed by: two anonymous referees