Abstract
Linear cases of Bragg–Hawthorne equation for steady axisymmetric incompressible ideal flows are systematically discussed. The equation is converted to a more convenient form in a spherical coordinate system. A new vorticity decomposition is derived. General solutions for 16 linear cases of the equation are obtained. These solutions can be specified to gain new analytical vortex flows, as examples in the paper demonstrate. A lot of well-known solutions like potential flow past a sphere, Hill's vortex with and without swirl, are included and extended in these solutions. Special relations between some vortex flows are also revealed when exploring or comparing related solutions.
Highlights
Bragg–Hawthorne equation[1] plays a central role in the study of axisymmetric steady flow of incompressible ideal fluids
An attempt is made to explore the linear cases in a systematic way and gain new explicit analytical solutions
The results are a series of general solutions, which, when specified, bring new analytical solutions of vortex flows
Summary
Bragg–Hawthorne equation[1] (or “B–H equation” in short) plays a central role in the study of axisymmetric steady flow of incompressible ideal fluids. The equation is rewritten in spherical coordinate system and the variable of polar angle is changed from h to cos h This gives a new form of the equation that is more convenient to solve in most cases. A series of assumptions for Bernoulli function and the azimuthal velocity that make the equation linear are listed. These assumptions lead to 16 combinations, each associated with a special linear case of the equation. A new decomposition of vorticity is derived This decomposition applies to all flows (that follow B–H equation) and helps us to understand the impacts of Bernoulli function and the azimuthal velocity to vorticity and to the flows.
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