Vorticity / Vortex flows

Helical vortex rings in the wake of a disk

(2026) 6, pp. 1-14. (pdf)

Thad S. Morton

Abstract:
The structure of the wake behind a circular disk is examined in numerical solutions for Re = 50, 100, and 120. A slowly reversing helical (swirling) streamline structure was found to exist in the steady numerical solutions at these flow speeds, which are generally regarded as steady and axisymmetric. The nested stream surfaces resemble Reeb-like foliations. The loss in fore-aft symmetry of the pressure field due to the viscous term may be associated with the azimuthal drift seen in the streamlines.

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How to obtain velocity fields from observed streamline patterns

(2009) 3, pp. 18-28. (pdf)

Thad S. Morton
Department of Integrated Engineering, Southern Utah University, Cedar City, UT 84720, USA

Abstract:
A flexible method is presented for making quantitative estimates of the velocity field within bounded regions of recirculating flow. Planar and axi-symmetric cases are considered. An expression for the vorticity field in the core of a vortex ring with a large elliptical cross section is given.

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An estimate of the circulation generated by a bluff body

(2008) 2, pp. 12-19. (pdf)

T. S. Morton
Aerospace & Biomedical Engineering, University of Tennessee Space Institute, 411 B.H. Goethert Parkway, Tullahoma, TN

Abstract:
A loss in circulation is sometimes cited in connection with bluff-body wakes as a result of comparing the circulation actually observed downstream with a well-known theoretical estimate of the total circulation generated by a cylinder. In an effort to better understand this reported loss in circulation, an alternative estimate of the circulation generated by a cylinder is derived by integrating the velocity on a closed loop containing the attached boundary layer. Predictions of the dimensionless circulation for a cylinder in crossflow are less than the previous theoretical estimate and agree with observed values. This suggests that the total circulation generated by bluff bodies may have been overestimated in the past, and that comparison of observed values with this overestimate is the origin of the perceived “loss” in circulation.

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A simplification of the vorticity equation and an extension of the vorticity persistence theorem to three dimensions

(2007) 1, pp. 21-29. (pdf)

T. S. Morton
Aerospace & Biomedical Engineering, University of Tennessee Space Institute, 411 B.H. Goethert Parkway, Tullahoma, TN

Abstract:
It has been known for more than a century that in two-dimensional (planar) Euler flow, vorticity is conserved along streamlines. In three-dimensions, however, no such result has been established, and this is primarily due to the vortex stretching term in the equation of motion. The vorticity persistence theorem is herein extended to three dimensions. It states that in any Euler flow, all components of the vorticity tensor of a streamline coordinate system that are normal to the streamline direction are conserved along streamlines. This extension is accomplished with the aid of a mathematical simplification of the vorticity equation derived for arbitrary coordinate systems. What remains of the nonlinear convective terms in the vorticity equation, after the mathematical simplification, is the Lie derivative of the vorticity tensor with respect to fluid velocity. A coordinate-independent temporal derivative is defined which, when set to zero, expresses either the continuity or vorticity equation (excluding the viscous term), depending upon the argument supplied to it.

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