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Vortex wake models

Vortex wake models represent a significant step forward in their potential ability, compared to the BEM methods, to capture the instantaneous loads of more complex and unsteady cases, but also a big increase in computational cost. In vortex wake models, the vorticity in the wake is modelled and used to determine the velocity field when and where needed. For VAWTs (and for HAWTs and rotorcraft), because the flow may be considered inviscid everywhere except for a thin layer trailing behind each blade, the vorticity can be represented as a thin sheet (or lines, or blobs, or points) of vorticity. The strength of vorticity in the wake is determined from the distribution of circulation along the span of each blade at the time of separation (the details of which are provided in section 4.3).
Vortex wake models can be broadly described as either fixed-wake and free-wake models. In fixed-wake models, a number of assumptions about the structure of the wake are made which make the models considerably less computationally demanding, at the expense of physical accuracy. In free-wake models, the wake is allowed to develop over time under its own influence and the influence of the structure, which results in much higher computational costs.
The use of vortex wake models is common for rotorcraft research [81–87] and although they have sometimes been used for HAWT research [88–91] they remain less popular than the BEM methods [92]. Vortex wake models have been applied to the analysis of VAWTs for almost as long as momentum models. Wilson applied a simple (by today’s standards) two-dimensional fixed-wake vortex model to the analysis of a Giromill5 as early as the late 1970s [93]. Wilson and Walker then developed a more advanced fixed-wake model capable of modelling curvedbladed Darrieus-type turbines that used a combination of vortex and momentum theory [94]. 5A Giromill is a straight-bladed VAWT with articulated blades that pitch to, in theory, allow maximal energy extraction.
The computational expense of these fixed-wake model was comparable to the pure momentum models and showed a reasonable capability to predict the overall performance of a turbine, but it is unclear whether the model offered much of an advantage over the pure momentum models. Coton et al. [95] developed a fixed-wake model for VAWT analysis with modifications to more accurately predict the tangential loads in unsteady conditions. Coton et al. demonstrated that their prescribed wake model might produce results comparable to a free-wake model (for the case tested), without the high computational burden.
Alongside the work being done on fixed-wake models at the time, Strickland et al. [96, 97] developed both two-dimensional and three-dimensional free-wake vortex models capable of modelling curved-bladed Darrieus-type VAWTs for analysis. A number of experimental tests were also performed and the numerical model compared favourably to the measurements [98, 99]. The vortex wake model lacked the advanced dynamic stall modelling capabilities described in this thesis (section 4.8), and the experimental tests were essentially two-dimensional only (straight-blades in a towing tank), but nonetheless, the comparison demonstrated the potential capabilities of free-wake vortex models for VAWTs. Modifications to the model were made later by Brownlee [100] in an effort to reduce the computational cost by assuming that the velocity of the wake vortices could be calculated at their creation time. Wilson et al. [101] also developed a two-dimensional free-vortex model that uses conformal mapping techniques to map the (unstalled) aerofoil into a rotating cylinder for analysis and showed good agreement with analytical predictions of unsteady motion. Vandenberghe and Dick [102] developed a free-wake vortex model for the analysis of VAWTs which, although only two-dimensional, did include a simple dynamic stall model (specifically, a modified Boeing-Vertol dynamic stall model) to account for some of the unsteady effects. Vandenberghe and Dick’s model differed from the other examples described here in that they used a vortex-in-cell approach. The vortices in the wake move in a Lagrangian manner, as in the other models, but the velocity induced by the wake is determined using a Eulerian grid. The vorticity is redistributed to the predefined grid nodes at each time-step. A hybrid Lagrangian-Eulerian method such as this can reduce the computational costs significantly while still retaining theoretically better unsteady modelling capabilities than the momentum methods. The main danger in this approach is in how the vorticity is redistributed from the vortices to the grid nodes. If not implemented correctly, it may lead to either numerical instabilities, or artificially introduced numerical diffusion.
The inherently unsteady nature of free-wake vortex models makes them theoretically much better suited to estimating instantaneous unsteady loads, but this must be weighed up against the fact that the computational cost of doing so is considerably greater than that of the momentum based models. On balance, because the unsteady aerodynamics were considered likely to have an effect on the aeroelastic behaviour of VAWTs, a free-wake vortex model was selected for use in the work described in this thesis. The details of the model implemented are described

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Other aerodynamic models

In addition to the momentum and vortex wake models discussed in the previous sections, several other methods have been used to model the aerodynamics of VAWTs, although they are generally much less common. These include cascade models, which are based on techniques typically applied to turbomachinery analysis. Modified versions of the cascade model developed by Hirsch and Mandal [103] for VAWT analysis were applied to an examination of dynamic stall and flow curvature effects by Mandal and Burton [104] and recently in an investigation on aerofoil design for small VAWTs by Islam et al. [47].
Navier-Stokes based computational fluid dynamics (CFD) modelling undoubtedly offers great potential for capturing an accurate representation of the aerodynamics. The greatest difficulty in modelling the aerodynamics of a structure as aerodynamically complex as a wind turbine using such methods, is however, the very high computational costs involved. Tchon and Paraschivoiu [105] used a two-dimensional Reynolds-averaged Navier-Stokes CFD model to simulate the flow around an aerofoil undergoing VAWT motion. Tchon and Paraschivoiu did not attempt to model the flow around the whole turbine, but rather modelled a single stationary aerofoil exposed to an flow that varied in direction in a manner approximating the variation in relative flow direction expected as a VAWT blade rotates. Allet et al. [106, 107] used a similar approach, but with an emphasis on using CFD to model the aerofoil undergoing VAWT motion as it experiences dynamic stall. Very recently, Howell et al. [56] used both two and three-dimensional CFD models to model a small VAWT and compared the results with wind tunnel tests. While such CFD work is interesting and may lead to some good insights, the very high computational cost of these CFD models limits their widespread use for the analysis of VAWTs.

1 Introduction
1.1 Introduction
1.2 The Problem
1.3 Summary of Approach .
1.4 Summary of Contributions
1.5 Structure of Thesis
2 Historical development of vertical-axis wind turbines
2.1 Introduction .
2.2 Historical wind energy development
2.3 Vertical-axis wind turbines
2.4 Aerodynamic modelling
2.5 Structural modelling
2.6 Aeroelastic modelling
3 Baseline turbine configuration
3.1 Introduction .
3.2 General configuration
3.3 Aerodynamic properties
3.4 Structural properties
4 Aerodynamic modelling
4.1 Introduction .
4.2 Overview of the aerodynamic simulation .
4.3 Vortex wake modelling
4.4 The wind field
4.5 The tower
4.6 Updating the wake state
4.8 Calculating the blade load coefficients .
4.9 Determination of convergence
4.10 Validation
5 Structural modelling
6 Aero-structural interface
7 Determination of required computational parameters
8 Case studies
9 Conclusions and recommendations
References