Abstract
In this paper, Beltrami vector fields in several orthogonal coordinate systems are obtained analytically and numerically. Specifically, axisymmetric incompressible inviscid steady state Beltrami (Trkalian) fluid flows are obtained with the motivation to model flows that have been hypothesized to occur in tornadic flows. The studied coordinate systems include those that appear amenable to modeling such flows: the cylindrical, spherical, paraboloidal, and prolate and oblate spheroidal systems. The usual Euler equations are reformulated using the Bragg-Hawthorne equation for the stream function of the flow, which is solved analytically or numerically in each coordinate system under the assumption of separability of variables. Many of the obtained flows are visualized via contour plots of their stream functions in the rz-plane. Finally, the results are combined to provide a qualitative quasi-static model for a progression of tornado-like flows that develop as swirl increases. The results in this paper are equally applicable in electromagnetics, where the equivalent concept is that of a force-free magnetic field.
Highlights
In fluid mechanics, Beltrami or helical flows are fluid flows in which the velocity and the vorticity of the fluid are parallel to each other at all points and all times
Axisymmetric incompressible inviscid steady state Beltrami (Trkalian) fluid flows are obtained with the motivation to model flows that have been hypothesized to occur in tornadic flows
Motivated by tornado-like flows shown in Figure 1, we focused on incompressible, steady, axisymmetric flows which allowed us to use a stream function formulation
Summary
Beltrami or helical flows are fluid flows in which the velocity and the vorticity (curl of velocity) of the fluid are parallel to each other at all points and all times. In this paper we will focus mainly on the hydrodynamics case, but many parallels can be drawn between the two, and results from one field can be applied in the other one An alternative to this approach is to study generalized Beltrami flows in which the curl of the cross product of the velocity and vorticity is zero, rather than the cross product itself [2] [3]. In this paper we analytically and numerically investigate axisymmetric incompressible inviscid steady state Beltrami (Trkalian, see below) flows with the goal of potentially developing other test cases for numerical models as well as possible models for tornadic or supercell flows. We explore several geometries characterized by various orthogonal coordinate systems and construct separable Beltrami solutions to the relevant equations in these systems.
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