The bidirectional vortex refers to the bipolar, coaxial, two-cell swirling motion that can be triggered, for example, in cyclone separators and some liquid rocket engines with tangential aft-end injectors. In this study, we derive an exact solution to describe the corresponding bulk motion in spherical geometry. Our approach invokes the assumptions of steady, incompressible, inviscid, rotational, and axisymmetric flow. Of the three possible types of similarity solutions that are shown to fulfill the momentum equation, only the second leads to a closed-form analytical expression that satisfies the boundary conditions for the bidirectional vortex in a straight cylinder. While the first type is incapable of satisfying the required conditions, its general form may be used to accommodate other physical settings. This case is illustrated in the context of inviscid flow over a sphere. The third type is more general and provides multiple solutions although it precludes a closed-form analytical outcome except for one case. The spherical bidirectional vortex is derived using separation of variables and the method of variation of parameters. The three-pronged analysis presented here increases our repertoire of general mean flow solutions that rarely appear in spherical geometry. It is hoped that these general forms will enable us to extend the current approach to other complex fluid motions that are simpler to capture using spherical coordinates. One such case corresponds to the analytical treatment of cyclonic flow in a conical chamber, a well known problem that remains unresolved.
|Titel||40th AIAA/ASME/SAE/ASEE Joint Propulsion Conference (Fort Lauderdale FL, USA, July 2004)|
|Status||Gepubliceerd - 2004|
Majdalani, J., Fang, D., & Rienstra, S. W. (2004). On the bidirectional vortex and other similarity solutions in spherical geometry. In 40th AIAA/ASME/SAE/ASEE Joint Propulsion Conference (Fort Lauderdale FL, USA, July 2004) (blz. 2004-3675-)