On Gaussian–Bessel Integrals in Diffusion‐Based Wind‐Turbine Wake Models

ABSTRACT

The diffusion wake model provides an analytical description of turbine‐wake evolution from a top‐hat profile in the near wake to a self‐similar Gaussian profile in the far wake. The main formulation of the model involves an integral containing the product of linear, Gaussian, and modified Bessel‐function terms. This integral is closely related to the Marcum ‐function and is commonly evaluated using series representations, limiting its analytical tractability. In this short communication, we develop a compact closed‐form approximation for this integral in terms of the error function. Comparisons with numerical quadrature show multi‐order‐of‐magnitude accuracy over the parameter range considered, with errors at least two orders of magnitude smaller than the corresponding function values. The approximation is then used to revisit the diffusion wake model and refine both its main formulation and its momentum‐conservation condition. In addition, the approximation is combined with recursive properties of the Marcum ‐function to obtain a compact representation of the generalized Marcum ‐function. The resulting formulation offers a simple and accurate alternative to series‐based evaluations while preserving the analytical structure of the original wake model.