An analytical solution for wind deficit decay behind a wind energy converter using momentum conservation validated by UAS data
Abstract. The wind deﬁcit behind a wind energy converter (WEC) results from a complex interaction of forces. Kinetic energy is removed from the atmosphere, but coherent turbulent structures prevent a swift compensation of momentum within the wake behind the WEC. A detailed description of the wake is beneﬁcial in meso-scale weather forecast (e.g. WRF models) and numerical simulations of wind wake deﬁcits. Especially in the near to intermediate wake (0.5−5 rotor diameters D), the dominating processes characterising the wake formation change along the wake. The conservation equation of momentum is used as a starting point to map the most important processes assuming the WEC operates at maximum efﬁciency in a neutral stratiﬁed boundary layer. The wake is divided into three different regions to accommodate the changing impact of atmospheric turbulence and the shear created by the WEC onto the wake. A differential equation that depicts the variable momentum transport into the wind deﬁcit along the wake is derived and solved analytically. Additionally, a numerical solution (Euler method) of the simpliﬁed momentum conservation equation is shown to provide a quality control and error estimate to the analytical model. The analytical solution is compared to conventional WEC wake models and in-situ wake measurements behind an Enercon E-112 converter, located in the Jade Wind Park near the North Sea coast in Germany, captured by the MASC-3 UAS (unmanned aircraft system) of the University of Tübingen. The obtained UAS data cover the distance from 0.5−5 D at hub height behind the nacelle. The analytical and numerical model are found to be in good agreement with the data of the three measurement ﬂights behind the WEC.
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