Beltrami flow structure in a diffuser. Quasi-cylindrical approximation

Abstract

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)].<p><strong>Received:</strong> 29 August 2011, <strong>Accepted:</strong> 29 February 2012; <strong>Edited by:</strong> J-P. Hulin; <strong>DOI:</strong> http://dx.doi.org/10.4279/PIP.040002</p><p><strong>Cite as:</strong> R González, R Page, A S Sartarelli, Papers in Physics 4, 040002 (2012)</p

Study of a Rotating Flow Trough a Contraction

“…En trabajos previos 1,2,3 hemos considerado flujos rotantes en un fluido ideal pasando de un cilindro a otro de mayor radio a través de una expansión axisimétrica. Abordamos en especial, un vórtice de Rankine 1 o un flujo formado por una rotación sólida más una translación uniforme 3 , aguas arriba. En esos casos el flujo rotacional aguas abajo de la expansión es un flujo de Beltrami en un sistema solidario con la rototranslación.…”