Exact Beltramian solutions for hemispherically bounded cyclonic flowfields

Abstract

This study focuses on the development of two analytical models that describe the wall-bounded cyclonic flowfield in a hemispherical domain. The closed-form solutions that we pursue are motivated by the need to characterize the swirling bidirectional motion engendered in an upper stage thrust chamber, namely, the VR35K-A VORTEX® engine, conceived and developed by Sierra Nevada Corporation. Our analysis proceeds from the Bragg–Hawthorne formulation, which is quite effective in the treatment of steady, inviscid, and axisymmetric flows. In this work, we show that two rotational solutions may be derived for particular specifications of the stagnation head and tangential angular momentum expressions that appear in the Bragg–Hawthorne equation. Then, with the parental streamfunctions in hand, other properties of interest are deduced and these include the main velocity and pressure variations, vorticities, crossflow velocities, extensional and shearing strain rates, virtual energy dissipation rates, and both axial and polar mantle distributions; the latter consist of pairs of rotating, non-translating interfacial layers, separating the so-called inner and outer bidirectional and bipolar regions, respectively. More specifically, two Beltramian solutions are identified with mantles that appear at 50% and 61.06% of the chamber radius, respectively. By matching the outlet radius of the chamber to the mantle location in the equatorial plane, the outflow is permitted to exit the chamber seamlessly. In both models, the axial and radial velocities vary linearly with the injection speed and a characteristic inflow parameter consisting of a geometric ratio of the inlet area and the chamber radius squared.

Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations

“…Note that ψ and C are not related a priori. For recent applications of the Bragg-Hawthorne equation, see, e.g., [17,22,23]. The Bragg-Hawthorne equation is an expression of the equation of motion for inviscid flow under steadiness assumption.…”

“…All of the calculations of the velocity field so far are at some fixed time. In order to determine the time-dependence of the solutions, we start from the fact that the time-dependent solution to the linear equation of motion (17) is generally expressed as a superposition of the single-mode solutions as…”

“…All of the calculations of the velocity field so far are at some fixed time. In order to determine the time-dependence of the solutions, we start from the fact that the time-dependent solution to the linear equation of motion (17) is generally expressed as a superposition of the single-mode solutions as ψ = r v(q, k, t)J 1 (qr)e ikz qdqdk + r G 0 (σ; σ )F θ (σ )dσ , v(q, k, t) = f (q, k)e −ν(q 2 +k 2 )(t+t 0 ) , (37) where σ ≡ (r, z, t) and v(q, k, t) is the Fourier-Hankel transform of v θ = ψ/r with the integration volume element dσ = rdrdzdt. The integration regions are such that 0 ≤…”

“…The marked characteristics of the Beltrami flow, the study of which was originated by Trkal [11] and Gromeka [12], is that the vorticity equations are apparently linear by virtue of the vanishing vector product of vorticity and velocity. In particular, inviscid Beltrami flows obeying the Euler equations have been of researchers' intense concern because of their close relevance to the flows with large Reynolds numbers that are expected to model the flows in meteorology [13,14] or engineering [15][16][17]. The so-called ABC flow that shows rich physical content in the Lagrangian dynamics provides a key to understanding chaotic or dynamo phenomena in fluids [18][19][20][21].…”

“…Steady axisymmetric inviscid Beltramian solutions in bounded or unbounded space have been found by focusing on the so-called Bragg-Hawthorne equation [17,22,23]. How the structure of the fundamental modes admitted by the linear equation varies by depending on the coordinate system chosen for solving the equation has been revealed in [22,24].…”