Login | Register

A Novel Formulation for Steady and Decaying Turbulent Line Vortices


A Novel Formulation for Steady and Decaying Turbulent Line Vortices

Georgios, Panagiotakakos (2017) A Novel Formulation for Steady and Decaying Turbulent Line Vortices. Masters thesis, Concordia University.

Text (application/pdf)
Panagiotakakos_MASc_S2017.pdf - Accepted Version
Available under License Spectrum Terms of Access.


Based on the n-family of laminar vortex formulation, a new generalized model applicable to the turbulent kind is presented. The self-similarity of the phenomenon allows, through the application of Vatistas and Aboelkassem (2006) simple variable transformation, to simulate its decay phase.
For the steady-state case, given the Vatistas et al. (1991) exponent n, the value of the turbulent intensity parameter , intrinsic in the azimuthal velocity formula, is found by fitting the analytical tangential velocity to various experimental profiles with different effective Reynolds numbers using the Least Square Error (LSE) method. Alike to the laminar n-family, n = 2 gives the smallest error and thus the best approximation. Also taking  to be constant or varying with the radius produces insignificant differences in the velocity profile. Thus, in order to close the system, the tangential velocity with n = 2 and a constant  that minimizes the error is inducted into the analysis. When  is plotted against the effective Reynolds number, a coherent relationship amongst the two emerges. An empirical equation, which connects the two properties, is then constructed. This gives the ability to researchers to approximate the velocity (tangential, axial and radial components) using only three parameters: the effective Reynolds number, the core radius, and the maximum tangential velocity.
Application of the abovementioned variable transformation to the steady turbulent vortex yields its corresponding decaying version. The validity of the model is tested for several laminar and turbulent cases. The tangential velocity decay of fixed wing aircraft wake, and rotating helicopter blade tip turbulent vortices, approximated using the new model provides more realistic results than the traditional circulation approach. The profiles of the last property, that is routinely used in aviation to define the hazard threshold in order to provide a safe aircraft separation distance in large airports, is found to be lacking in representing the real cases of diminishing vortices. The previous lies on the fact that the assumed flattening of the circulation curve at large radii, applicable to laminar cases it is not true when the vortex is turbulent. Consequently, the prescribed value of the radius (e.g. 7 times the core radius) to represent the circulation at “infinity” proposed by Squire (1965) and Iversen (1976), implemented also in numerous other models like Burnham and Hallock (1982) and Proctor (2000), must be reconsidered. Future work should focus on the definition of the hazard threshold based on the tangential velocity instead of its circulation signature.

Divisions:Concordia University > Gina Cody School of Engineering and Computer Science > Mechanical and Industrial Engineering
Item Type:Thesis (Masters)
Authors:Georgios, Panagiotakakos
Institution:Concordia University
Degree Name:M.A. Sc.
Program:Mechanical Engineering
Date:12 March 2017
Thesis Supervisor(s):Vatistas, Georgios
ID Code:982445
Deposited On:09 Jun 2017 14:46
Last Modified:18 Jan 2018 17:55


Aboelkassem Y.andVatistas,G.H. (2007) “New Model for Compressible Vortices,” J. Fluid Eng. Vol. 29, pp.1073-1079.
Ansari S. A., Zbikowski, R., and Knowles, A. (2006) “Non-Linear Unsteady Aerodynamic Model for Insect-Like Flapping Wings in the Hover. Part 2: Implementation and Validation,” Journal of Aerospace Engineering, Vol. 220, No. 3, pp. 169–186.
Antonini E. G. A., Bedon, G., De Betta, S., Michelini, L., Marco Raciti Castelli, M. R., and Benini, E. (2015) “Innovative Discrete-Vortex Model for Dynamic Stall Simulations,” AIAA Journal, Vol. 53, No. 2, pp. 479–485.
Bennett T. J. (1988) “Vortex Coalescence and Decay,” Ph.D. dissertation, Department of Civil and Environmental Engineering, Washington State University, Pullman, Wash. USA.
Bhagwat M. J., and Leishman, J. G. (2000) “Correlation of Helicopter Tip Vortex Measurements,” AIAA Journal, Vol. 38, No. 2, 2000, pp. 301–308.
Bhagwat M. J., and Leishman, J. G. (2002) “Viscous Vortex Core Models for
Free-Vortex Wake Calculations,” Proceedings of the 58th Annual Forum of the American Helicopter Society International, 11–13, June 2002.
Bhagwat M. J., and Leishman, J. G. (2000)b “Measurements of Bound and Wake Circulation on a Helicopter Rotor,” Journal of Aircraft, Vol. 37, No. 2, 2000, pp. 227–234.
Boltzmann L., “Zur Integration der Diffusionsgleichung bei Variabeln Diffusions-coefficienten,” Annalen der Physik, Vol. 53, 1894, pp. 959–964.
Brix S., Neuwerth, G., and Jacob, D., (2000) “The Inlet-Vortex System of Jet Engines Operating Near the Ground,” AIAA Paper 2000-3998.
Burnham D.C., Hallock, J.N. (2013) “Decay Characteristics of Wake Vortices from Jet Transport Aircraft,” Journal of aircraft, Vol. 50, No. 1, pp. 82-87.
Burnham D.C., Hallock, J.N. (1982) “Chicago Monostatic Acoustic Vortex Sensing System,” U.S. Department of Transportation, DOT-TSC-FAA-79-103, 206 p.
Burnham D. C., Hallock, J. N., Tombach, I. H., Brashears, M. R. and Barber, M. R. (1978) “Ground-Based Measurements of the Wake Vortex Characteristics of a B-747 Aircraft in Various Configurations,” Dept. of Transportation, Transportation Systems Center Rept. FAA-RD-78-146, Cambridge, MA.
Burgers J. M. (1948) ‘‘A Mathematical Model Illustrating the Theory of Turbulence,’’ Advances in Applied Mechanics, Vol. 1, pp. 171– 199.
Caradonna F. X., and Tung, C. (1981) “Experimental and Analytical Studies of a Model Helicopter Rotor,” Vertica, Vol. 5, pp. 149–161.

Cotel A. J., and Breidenthal, R. E. (1999) “Turbulence inside a Vortex,” Physics of Fluids, Vol. 11, No. 10, pp. 3026–3029.
Davenport W. J., Rife, M. C., Liapis, S. I., and Follin, G. J. (1996) “The Structure and Development of a Wing-tip Vortex,” Journal of Fluid Mechanics, Vol. 312, pp. 67–106.
Delisi D. P., Greene, G. C., Robins, R. E., Vicroy, D. C., and Wang, F. Y. (2003) “Aircraft Wake Vortex Core Size Measurements,” 21st Applied Aerodynamics Conference, AIAA Paper 2003-3811.
Dosanjh D. S., Gasparek, E. P. and Eskinazi, S. (1962) “The Decay of a Viscous Trailing Vortex,” Aeronautical Quarterly, 13 (2), pp. 167–188.
Grifiths R. W., and Hopfinger, E. J. (1987) “Coalescing of Geostrophic Vortices,” Journal of Fluid Mechanics, Vol. 178, pp. 73–97.
Han Y. O., Leishman, J. G. and Coyne, A. J. (1997) “On the Turbulent Structure of a Tip Vortex Generated by a Rotor,” AIAA Journal, Vol. 35, No. 3, March 1997,pp. 477–485.
Haw P. (1969) “The Analogy Between Streamline Curvature and Buoyancy in Turbulent Shear Flows,” Journal of Fluid Mechanics, Vol. 36, pp. 177–191.
Hinton D. A. and Tatnall C. R. (1997) “A Candidate Wake Vortex Strength Definition for Application to the NASA Aircraft Vortex Spacing System (AVOSS),” NASA Technical Memorandum 110343.
Holzapfel A. Hofbauer, T., Gerz, T., and Schumann, U. (2001) “Aircraft Wake Vortex Evolution and Decay in Idealized and Real Environtments: Methodologies, Benefits and Limitations,” Proceedings of the Euromech Colloquium, 2001.
Holforty L. W. (2003) “Flight-Dec Display of Neighboring Aircraft Vortices”, PhD. Thesis, Department of Aeronautics and Astronautics, Stanford University, USA.
Hoffman E. R., and Joubert P. N. (1963) “Turbulent Line Vortices,” Journal of Fluid Mechanics, Vol. 16, 1963, pp. 395–411.
Iversen J. D. (1973) “Correlation of Turbulent Trailing Vortex Decay Data,” Journal of Aircraft, Vol. 13, No. 5, pp. 338-342.
Kaufmann W. (1962) “̈Über die Ausbreitung kreiszylindrischer Wirbel in zähen (viskosen) Flüssigkeiten.” Ingenieur-Arch. Vol. 31, No1, PP. 1- 9.
Koval P.V. and Michaelov, P. S. (1972) “Velocity and Pressure Distributions of Liquid in a Swirl Chamber.” Teploenergetica, vol.19, no.2, pp. 25-28.
Kecskemety K. M., and McNamara, J. J., (2011) “Influence of Wake Effects and Inflow Turbulence on Wind Turbine Loads,” AIAA Journal, Vol. 49, No. 11, pp. 2564–2576.
Lamb H., (1932)“Hydrodynamics,” 6th ed., Cambridge Univ. Press, New York, pp. 591-592.
Leishman J. G., (1998) “Measurements of the Aperiodic Wake of a Hovering Rotor,” Experiments in Fluids, Vol. 25, No. 4, pp. 352–361.
Lundgren T. S. (1982) “Strained spiral vortex model for turbulent fine structure,” Physics of Fluids, Vol. 25, No. 12, pp. 2193-2203.
Martin P., and Leishman, J. G., (2002) “Trailing Vortex Measurements in the Wake of a Hovering Rotor with Various Tip Shapes,” Proceedings of the 58th Annual Forum of the American Helicopter Society International, Montreal Canada, June 11–13, 2002.
Martin P. B., Pugliese, G., and Leishman, J. G., (2001) “High Resolution Trailing Vortex Measurements in the Wake of a Hovering Rotor,” American Helicopter Society 57th Annual National Forum, Washington, DC, May 9–11.
Mahalingam R., and Komerath, N. M., (1998) “Measurements of the Near Wake of a Rotor in Forward Flight,” AIAA Paper 98-0692.
McAlister K. W., (1996) “Measurements in the Near Wake of a Hovering Rotor,” AIAA Paper 96-1958.
McCormick B. W., Tangler, J. L., and Sherrieb, H. E., (1968) “Structure of Trailing Vortices,” Journal of Aircraft, Vol. 5, No. 3, pp. 260–267.
Meunie, P., and Villermaux, E., (2003) “How Vortices Mix,” Journal of Fluid Mechanics, Vol. 476, pp. 213–222.
Murphy J. P., and MacManus, D. G. (2011) “Ground Vortex Aerodynamics Under Crosswind Conditions,” Experiments in Fluids, Vol. 50, No. 1, 2011, pp. 109–124.
Murphy J. P., and MacManus, D. G. (2011) “Inlet Ground Vortex Aero- 
dynamics Under Headwind Conditions,” Aerospace Science and Technology, Vol. 15, No. 3, pp. 207–215.
Newman B. G. (1959) “Flow in a Viscous Trailing Vortex,” Aeronautical Quarterly, 10 (2), pp. 149–162.
Osseen C. W. (1912) “Uber Wirbelbewegune in Einer Reiben-den Flussigkeit,” Arkiv för Matematik, Astronomi och Fysik, Vol. 7, pp. 14–21.
Proctor F.H., Hamilton, D.W. and Han, J. (2000) "Wake Vortex Transport and Decay in Ground Effect: Vortex Linking with the Ground", AIAA Paper 2000-0757.
Ramasamy M. and Leishman, J. G. (2007) “A Reynolds Number-Based Blade Tip Vortex Model,” Journal of the American Helicopter Society, Vol. 52, Issue 3, pp. 214–223.
Ramesh K., Murua, J., and Gopalarathnam, A., “Limit-Cycle Oscillations in Unsteady Flows Dominated by Intermittent Leading-Edge Vortex Shedding,” Journal of Fluids and Structures, Vol. 55, May 2015, pp. 84–105.
Rankine W. J. M. (1858) “Manual of Applied Mechanics,” C. Griffen Co., London, 1858, pp. 576–578.
Rossow V. J. (2006) “Origin of Exponential Solution for Laminar Decay of Isolated Vortex,“ Journal of Aircraft, Vol. 43, No. 3, pp. 709-712.
Scully M. P. (1975) “Computation of Helicopter Rotor Wake Geometry and Its Influence on Rotor Harmonic Airloads,” Aeroelastic and Structures Research Lab., Massachusetts Inst. of Technology TR ASRL TR-178-1, Cambridge, MA, March.
Snedeker R. S. (1972) “Effect of Air Injection on the Torque Produced by a Trailing Vortex,” Journal of Aircraft, Vol. 9, No. 9, pp. 682–684. 

Squire H.B. (1965) "The Growth of a Vortex in Turbulent Flow," Aeronautical Quarterly, Vol. 16, Part 3, August 1965, pp. 302-306.
Sullivan R. D. (1959) "A Two-Cell Vortex Solution of the Navier-Stokes Equations," Journal of the Aerospace Sciences, Vol. 26, pp. 767-768.
Takahashi R. K., and McAlister, K. W. (1987) “Preliminary Study of a Wing- Tip Vortex Using Laser Velocimetry,” NASA TM-88343.

Tao S., Jianfeng, T., and Haowen, W., (2013) “Investigation of Rotor Control System Loads,” Chinese Journal of Aeronautics, Vol. 26, No. 5, pp. 1114–1124.
Tung C., Pucci, S. L., Caradonna, F. X., and Morse, H. A., “The Structure of Trailing Vortices Generated by Model Helicopter Rotor Blades,” NASATM 81316, 1981.
Vatistas G. H., (2006) “Simple Model for Turbulent Tip Vortices,” Journal of Aircraft, Vol. 43, No. 5, pp. 1577–1579.
Vatistas H. G., and Aboelkassem, Y., (2006) “Space-Time Analogy of Self-Similar Intense Vortices,” AIAA Journal, Vol.44, No. 4, pp. 912-917.
Vatistas, G. H. (1998) “New Model for Intense Self-Similar Vortices.” Journal of Propulsion and Power, Vol. 14, No. 4, pp. 462–469.
Vatistas G. H., Kozel, V., and Mih, W. (1991) “A Simpler Model for Concentrated Vortices,” Experiments in Fluids, Vol. 11, No. 1, pp. 73–76.
Vatistas G.H., Panagiotakakos, G. D..and Manikis, F. I. (2015) “Extension of the n-Vortex Model to Approximate the Effects of Turbulence,” Journal of Aircraft, Vol. 52, No. 5, pp. 1721-1725.
Vatistas G.H. (2004) “The fundamental properties on the n = 2 vortex model.” Trans. Canadian Society of Mechanical Engineers, Vol. 28, pp. 43-58.
Wang Y. and Jiang, C. (2010) “Investigation of the Surface Vortex in a Spillway Tunnel Intake,” Tsinghua Science and Technology, Vol. 15, 
No. 5, pp. 561–565.
Zheng T. H., Vatistas, G.H., and Povitsky, A. (2007) “Sound Generation by a Street of Vortices in a Non uniform Flow,” Physics of Fluids, Vol. 19, 03710.
All items in Spectrum are protected by copyright, with all rights reserved. The use of items is governed by Spectrum's terms of access.

Repository Staff Only: item control page

Downloads per month over past year

Back to top Back to top