Summary: In [Topology 9, 153–154 (1970; Zbl 0177.52103)], D. Tischler proved that a manifold admitting a smooth non-vanishing and closed one-form fibres over a circle. More generally, a manifold admitting \(k\)-independent closed one-form fibres over a torus \(T^k\). In this article, we explain a version of this construction for manifolds with boundary using the techniques of \(b\)-calculus [R. B. Melrose, The Atiyah-Patodi-Singer index theorem. Wellesley, MA: A. K. Peters, Ltd. (1993; Zbl 0796.58050); V. Guillemin et al., Adv. Math. 264, 864–896 (2014; Zbl 1296.53159)]. We explore new applications of this idea to fluid dynamics and more concretely in the study of stationary solutions of the Euler equations. In the study of Euler flows on manifolds, two dichotomic situations appear. For the first one, in which the Bernoulli function is not constant, we provide a new proof of Arnold’s structure theorem and describe \(b\)-symplectic structures on some of the singular sets of the Bernoulli function. When the Bernoulli function is constant, a correspondence between contact structures with singularities [the second author and C. Oms, “Contact structures with singularities”, Preprint, arxiv:1806.05638] and what we call \(b\)-Beltrami fields is established, thus mimicking the classical correspondence between Beltrami fields and contact structures (see for instance [J. Etnyre and R. Ghrist, Trans. Am. Math. Soc. 352, No. 12, 5781–5794 (2000; Zbl 0960.76020)]). These results provide a new technique to analyse the geometry of steady fluid flows on non-compact manifolds with cylindrical ends.