  Arrays of Vertical Axis Wind Turbines (VAWTs) can achieve larger power
generation per land area than horizontal axis turbines farms, due to the
positive synergy between VATs in close proximity. Theoretical wake models
enable the reliable design of the array layout that maximises the energy
output, which need to depict the driving wake dynamics. VAWTs generate a highly
complex wake that evolves according to two governing length-scales, namely the
turbine rotor's diameter and height which define a rectangular shape of the
wake cross-section, and feature distinct wake expansion rates. This paper
presents analytical VAWT wake models that account for an asymmetric
distribution of such wake expansion adopting a top-hat and Gaussian velocity
deficit distribution. Our proposed analytical Gaussian model leads to an
enhanced initial wake expansion prediction with the wake width ($\varepsilon$)
behind the rotor equal to $(\beta/4 \pi)^{1/2}$ with $\beta$ being the ratio of
initial wake area to the VAWT's frontal area, which addresses the limitations
of previous models that under-predicted the wake onset area. Velocity deficit
predictions are calculated in a series of numerical benchmarks consisting of a
single and an array of four in-line vertical axis wind turbines. In comparisons
with field data and large-eddy simulations, our models provide a good accuracy
to represent the mean wake distribution, maximum velocity deficit, and momentum
thickness, with the Gaussian model attaining the best predictions.These models
will aid to drive the design of VAT arrays and accelerate this technology.
