Helical Models of the Bidirectional Vortex in a Conical Geometry

This dissertation represents the descriptive and analytical breakdown of two new fluid dynamics solutions for vortex motion. Both solutions model the bidirectional vortex within a conical geometry. The first explored solution satisfies a simple Beltramian characteristic, where the Lamb vector is identically zero. The second solution is of the generalized Beltramian type, which fulfills the condition that the curl of the Lamb vector is equal to zero. The two Beltramian solutions describe the axisymmetric, double helical motion often found in industrial cyclone separators. Other applications include cone-shaped, vortex-driven combustion chambers and the swirling flow through conical devices. Both solutions are derived from first principles and Euler’s equations of motion which showcase the stream function-vorticity relation and ultimately transforms into the Bragg-Hawthorne formulation. The Bragg-Hawthorne equation allows for various implementations of the Bernoulli and swirl functions. The angular momentum equation includes the source term for the Beltramian solution. On the other hand, the Bernoulli relation drives the generalized Beltramian model. Appropriate boundary conditions and assumptions reduce the governing partial differential equation to an ordinary differential equation which is then solved by a separation of variables approach. Resulting velocity, vorticity, and pressure variables are discussed and graphed. The tangential and axial velocities are compared to two experimental and numerical cyclone separator cases. Other features of the conical flow field such as the conical swirl number and dual mantle locations are also explored. The inviscid, incompressible, and rotational models ultimately lay the framework for complementary solutions derived from the Bragg-Hawthorne equation or similar formulations