New Model for Compressible Vortices

[+] Author and Article Information
Yasser Aboelkassem1

Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, Quebec, H3A-2K6, Canadayasser.aboelkassem@mail.mcgill.ca

Georgios H. Vatistas

Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec H3G 1M8, Canadavatistas@encs.concordia.ca


Corresponding author.

J. Fluids Eng 129(8), 1073-1079 (Feb 26, 2007) (7 pages) doi:10.1115/1.2746897 History: Received March 30, 2006; Revised February 26, 2007

A new analytical solution for self-similar compressible vortices is derived in this paper. Based on the previous incompressible formulation of intense vortices, we derived a theoretical model that includes density and temperature variations. The governing equations are simplified assuming strong vortex conditions. Part of the hydrodynamic problem (mass and momentum) is shown to be analogous to the incompressible kind and as such the velocity is obtained through a straightforward variable transformation. Since all the velocity components are bounded in the radial direction, the density and pressure are then determined by standard numerical integration without the usual stringent simplification for the radial velocity. While compressibility is shown not to affect the tangential velocity, it influences only the meridional flow (radial and axial velocities). The temperature, pressure, and density are found to decrease along the converging flow direction. The traditional homentropic flow hypothesis, often employed in vortex stability and optical studies, is shown to undervalue the density and greatly overestimate the temperature. Comparable to vorticity diffusion balance for the incompressible case, the incoming flow carries the required energy to offset the contributions of conduction, viscous dissipation, and material expansion, thus keeping the temperature steady. This model is general and can be used to obtain a compressible version for all classical previous incompressible analysis from the literature such as Rankine, Burgers, Taylor, and Sullivan vortices.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

The problem coordinate system

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Figure 2

Correlations of the n=2 vortex with the experimental data

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Figure 3

The magnitude of the different terms in the energy equation 15

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Figure 4

The static pressure (a), density (b), and temperature (c). (The isentropic curves in (a)–(c) are calculated using Eq. 16.

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Figure 6

Entropy increase along the flow direction

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Figure 5

The radial and axial velocity components of the meridional flow



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