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We do this by generalizing the spectral approach of Laplacian Eigenfunctions, resulting in what we call\n            <jats:italic>spiral-spectral<\/jats:italic>\n            fluid simulations. Starting with a set of divergence-free analytical bases for polar and spherical coordinates, we show that their singularities can be removed by introducing a set of carefully selected enrichment functions. Orthogonality is established at minimal cost, viscosity is supported analytically, and we specifically design basis functions that support scalable FFT-based reconstructions. Additionally, we present an efficient way of computing all the necessary advection tensors. Our approach applies to both three-dimensional flows as well as their surface-based, codimensional variants. 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