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Beltrami flow structure in a diffuser. Quasi-cylindrical approximation

We determine the flow structure in an axisymmetric diffuser or expansion region connecting two cylindrical pipes when the inlet flow is a solid body rotation with a uniform axial flow of speeds Omega and U, respectively. A quasi-cylindrical approximation is made in order to solve the steady Euler equation, mainly the Bragg-Hawthorne equation. As in our previous work on the cylindrical region downstream [R González et al., Phys. Fluids 20, 24106 (2008); R. González et al., Phys. Fluids 22, 74102 (2010), R González et al., J. Phys.: Conf. Ser. 296, 012024 (2011)], the steady flow in the transition region shows a Beltrami flow structure. The Beltrami flow is defined as a field v_B that satisfies omega_B=nabla v_B= gamma v_B, with gamma = constant. We say that the flow has a Beltrami flow structure when it can be put in the form v = U e_z + Omega r e_theta + v_B, being U and Omega constants, i.e it is the superposition of a solid body rotation and translation with a Beltrami one. Therefore, those findings about flow stability hold. The quasi-cylindrical solutions do not branch off and the results do not depend on the chosen transition profile in view of the boundary conditions considered. By comparing this with our earliest work, we relate the critical Rossby number vartheta_cs (stagnation) to the corresponding one at the fold vartheta_cf [J. D. Buntine et al., Proc. R. Soc. Lond. A 449, 139 (1995)].