Abstract
In this work, an exact Eulerian model is used to describe the steady-state motion of a bidirectional vortex in a conical chamber. This particular model is applicable to idealized representations of cyclone separators and liquid rocket engines with slowly expanding chamber cross-sections. The corresponding bulk motion is assumed to be non-reactive, rotational, inviscid and incompressible. Then, following Bloor & Ingham (J. Fluid Mech., vol. 178, 1987, pp. 507–519), the spherical Bragg–Hawthorne equation is used to construct a mathematical model that connects the solution to the swirl number and the cone divergence angle. Consequently, a self-similar formulation is obtained independently of the cone’s finite body length. This enables us to characterize the problem using closed-form approximations of the principal flow variables. Among the cyclonic parameters of interest, the mantle divergence angle and the maximum cross-flow velocity are obtained explicitly. The mantle consists of a spinning cone that separates the circumferential inflow region from the central outflow. This interfacial layer bisects the fluid domain at approximately 60 per cent of the cone’s divergence half-angle. Its accurate determination is proven asymptotically using two different criteria, one being preferred by experimentalists. Finally, recognizing that the flow in question is of the Beltramian type, results are systematically described over a range of cone angles and spatial locations in both spherical and cylindrical coordinates; they are also compared to available experimental and numerical data.
Highlights
Cyclonic flow field modelling is presently undergoing an era of renewed interest, especially in advanced propulsion-related combustion devices, where swirl-dominated cyclonic motions have proven to be beneficial due to their self-cooling properties, enhanced stability and elevated efficiencies
Several prototypical engines driven by liquid propellant or hybrid fuel combustion are under development today following the bidirectional vortex notion introduced by Knuth and co-workers in 1996
Beltrami flow in a conical cyclone examples include those concerned with swirl-driven hybrid rocket engines by Knuth et al (1996, 2002) and those associated with liquid–liquid thrust chambers by Sauer et al (2002), Chiaverini et al (2003) and Majdalani (2007, 2012)
Summary
Cyclonic flow field modelling is presently undergoing an era of renewed interest, especially in advanced propulsion-related combustion devices, where swirl-dominated cyclonic motions have proven to be beneficial due to their self-cooling properties, enhanced stability and elevated efficiencies. Their work was further extended by Derksen & van den Akker (2000), who used large eddy simulations to predict the fundamental flow features evolving in a reverse-flow cyclone separator Their investigation included the precession patterns of the core vortex and its stabilizing role. Despite its inviscid character and limitations to small cone angles and simplistic boundary conditions, the Bloor–Ingham solution is repeatedly shown to exhibit similar characteristics to the flow simulated numerically by Hsieh & Rajamani (1991), Hoekstra et al (1999) and Derksen & van den Akker (2000) For this reason, the Bloor–Ingham model will be carefully revisited and extended in the context of a conically shaped cyclonic chamber. The salient flow features in this problem will be discussed and compared whenever possible to existing models in the literature
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