Parabolized Eddy Viscosity Models for Wind Turbine Wake Modelling

Abstract

As the share of wind energy in the global electricity mix increases, it becomes more important to accurately predict wind turbine wakes in order to optimise wind farm layouts and to reduce fatigue loads on downstream turbines. Wake models based on the Reynolds-averaged Navier–Stokes equations offer a good balance between accuracy and computational cost, but their performance depends strongly on the turbulence closure used. The standard k–ε model is widely used in wind energy, but it is known to overpredict the eddy viscosity in the near wake. This leads to an overestimation of turbulent mixing and wake recovery. To address this, researchers have proposed extended closures, which have only been validated in fully elliptic three-dimensional RANS solvers. The Forced Ainslie Wake Model has improved the physical consistency of parabolic wake modelling by reintroducing part of the actuator disk force. Its turbulence closure is currently limited to a constant eddy viscosity. A transport-based turbulence model that is compatible with the parabolic marching scheme of the FAWM is therefore needed.

This thesis develops and validates a parabolised, axisymmetric k–ε turbulence model that can be coupled to the Forced Ainslie Wake Model. Three turbulence closures are implemented: the standard k–ε model and two of its extensions. The k–ε–fP model limits the eddy viscosity in regions of high shear through a variable C*μ derived from the nonlinear eddy viscosity model framework. The extended k–ε model with a sink term Sk accounts for the extraction of turbulent kinetic energy by the actuator disk. The transport equations for k and ε are discretised on a staggered finite volume grid using a second-order upwind scheme for the convective terms and central differencing for the diffusive terms. The turbulence model is coupled with the FAWM through a Picard iteration scheme that exchanges velocity and eddy viscosity at each axial station until convergence. The implementation is verified through the decay of isotropic turbulence, the Method of Manufactured Solutions, and a self-similar co-flowing jet simulation. In all three tests, the solver achieves the formal order of convergence. This confirms the correct implementation of the discretisation and the coupling between the transport equations.

During the validation against LES data for a DTU 10 MW reference wind turbine across four operating conditions, numerical stability problems were encountered. In the original model formulation, only 5 out of 12 simulations converged. Some of the converged cases produced non-physical values of the turbulent kinetic energy. The source of the instabilities was traced to the axial gradient of the radial velocity, ∂V/∂x, which contributes to the strain-rate invariant and therefore the production term. Because the radial velocity is derived from the axial velocity through the continuity equation, ∂V/∂x depends on the second axial derivative of U. Any irregularity in the actuator disk forcing is transferred to the axial velocity through the momentum equation. In a parabolic solver, the axial diffusion and pressure terms that would normally dampen these gradients are absent, leading to an overestimation of ∂V/∂x. Neglecting ∂V/∂x in the computation of the production term restored convergence in 11 out of 12 cases. This also keeps all predicted values within physical bounds.

With the modified formulation, the three turbulence closures were compared against the LES reference data. For the axial velocity, the k–ε–fP model performs best in the near wake, with MAPE values between 1 and 4%, while the k–ε–Sk model produces the lowest errors in the far wake. The standard k–ε model has the largest velocity errors across all configurations. The velocity errors increase with increasing thrust coefficient and decreasing wind speed. For the turbulent kinetic energy, all three models overpredict the LES values by a large margin, with full-wake MAPE values ranging from 50 to 126%. The k–ε–Sk model produces the lowest TKE errors in every configuration, while the standard model and k–ε–fP model have comparable errors.

In summary, this thesis demonstrates that parabolised turbulence models can be coupled with the parabolic FAWM solver. It also shows that the extended closures improve predictions compared to the standard k–ε model. However, the original model formulation with ∂V/∂x included in the production term is not reliably stable, and the presented results were obtained with a modified formulation. No single closure is best for all flow variables simultaneously. Further work is needed to resolve the stability issue of the original formulation.