(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.
References:
Aref, H., “Stirring by chaotic advection,” Journal of Fluid Mechanics 143 (1984) pp. 1-21.
Arnold, V. I., “Sur la topologie des écoulements stationnaires des fluides parfaits,” Comptes Rendus de l’Académie des Sciences 261 (1965) pp. 17-20.
Arnold, V. I., “On the topology of three-dimensional steady flows of an ideal fluid,” Journal of Applied Mathematics and Mechanics (English translation of Prikladnaya Matematika i Mekhanika) 30 (1966) pp. 223–226.
Auguste, F., Fabre, D., & Magnaudet, J., “Bifurcations in the wake of a thick circular disk,” Theoretical and Computational Fluid Dynamics 24 (2010) pp. 305-313.
Berger, E., Scholz, D., & Schumm, M., “Coherent vortex structures in the wake of a sphere and a circular disk at rest and under forced vibrations,” Journal of Fluids and Structures 4 (1990) pp. 231-257.
Calvert, J. R., “Experiment on the flow past an inclined disk,” Journal of Fluid Mechanics 29 (1967) pp. 691-703.
Camacho, C. & Neto, A. L., Geometric Theory of Foliations, Birkhäuser Boston (1984).
Candel, A. & Conlon, L., Foliations I, American Mathematical Society, Providence, Rhode Island (2000).
Fabre, D., Auguste, F., & Magnaudet, J., “Bifurcations and symmetry breaking in the wake of axisymmetric bodies,” Physics of Fluids 20 (2008) pp. 051702-1-4.
Fernandes, P. C., Risso, F., Ern, P., & Magnaudet, J., “Oscillatory motion and wake instability of freely rising axisymmetric bodies,” Journal of Fluid Mechanics 573 (2007) pp. 479-502.
Fuchs, H. V., Mercker, E., & Michel, U., “Large-scale coherent structures in the wake of axisymmetric bodies,” Journal of Fluid Mechanics 93 (1979) pp. 185-207.
Hadamard, J., “Mouvement permanent lent d’une sphere liquide et visqueuse dans un liquidemvisqueux,” Comptes Rendus 152 (1911) pp. 1735-1738.
Hill, M. J. M., “On a Spherical Vortex,” Philosophical Transactions of the Royal Society of London A 185 (1894) pp. 213-245.
Hwang, H. H. C. & Baldwin, L. V., “Decay of Turbulence in Axisymmetric Wakes,” ASME Journal of Basic Engineering, D 88 (1966) pp. 261-268.
Jeffreys, H., “The Wake in Fluid Flow Past a Solid,” Proceedings of the Royal Society (London) A 128 (1930) pp. 376-393.
Johnson, T. A. & Patel, V. C., “Flow past a sphere up to a Reynolds number of 300,” Journal of Fluid Mechanics 378 (1999) pp. 19-70.
Lawson, Jr., H. B., “Foliations,” Bulletin of the American Mathematical Society 80 (1974) pp. 369-418.
Lee, S. I. & Bearman, P. W., “An experimental investigation of the wake structure behind a disk,” Journal of Fluids and Structures 6 (1992) pp. 437-450.
Levi, E., “A Universal Strouhal Law,” Journal of Engineering Mechanics, ASCE 109 (1983) pp. 718-727.
Llewellyn Smith, S. G. & Ford, R., “Three-dimensional acoustic scattering by vortical flows. II. Axisymmetric scattering by Hill’s spherical vortex,” Physics of Fluids 13 (2001) pp. 2890-2900.
Lückoff, F., Sieber, M., Paschereit, C. O., & Oberleithner, K., “Characterization of Different Actuator Designs for the Control of the Precessing Vortex Core in a Swirl-Stabilized Combustor,” Journal of Engineering for Gas Turbines and Power 140 (2018) pp. 041503-1-10.
Magarvey, R. H. & Bishop, R. L., “Transition ranges for three-dimensional wakes,” Canadian Journal of Physics 39 (1961) pp. 1418-1422.
Marshall, D. & Stanton, T. E., “On the Eddy System in the Wake of Flat Circular Plates in Three Dimensional Flow,” Proceedings of the Royal Society of London A 130 (1931) pp. 295-301 (17 pages including plates).
Miau, J. J., Leu, T. S., Liu, T. W., & Chou, J. H., “On vortex shedding behind a circular disk,” Experiments in Fluids 23 (1997) pp. 225-233.
Mishra, P., Patel, S.A., Trivedi, M., & Chhabra, R.P., “Effect of Power-Law Fluid Behaviour on Nusselt Number of a Circular Disk in the Forced Convection Regime,” Journal of Heat Transfer 141 (2019) pp. 041701-1-8.
Morton, T. S., “The velocity field within a vortex ring with a large elliptical cross-section,” Journal of Fluid Mechanics 503 (2004) pp. 247-271.
Morton, T. S., “A correlation between drag and an integral property of the wake vortex,” Journal of Scientific and Mathematical Research 1 (2007) pp. 2-20.
Morton, T. S., “A parameter for a family of steady vortex ring cores,” European Journal of Mechanics – B/Fluids 56 (2016) pp. 66-70.
Natarajan, R. & Acrivos, A., “The instability of the steady flow past spheres and disks,” Journal of Fluid Mechanics 254 (1993) pp. 323-344.
Nitin, S. & Chhabra, R.P., “Wall effects in two-dimensional axisymmetric flow over a circular disk oriented normal to flow in a cylindrical tube,” The Canadian Journal of Chemical Engineering 83 (2005) pp. 450-457.
O’Brien, V., “Steady spheroidal vortices – More exact solutions to the Navier-Stokes equation,” Quarterly of Applied Mathematics 19 (1961) pp. 163-168.
Ottino, J. M., The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge University Press, Cambridge (1989).
Palies, P., Durox, D., Schuller, T., Morenton, P., & Candel, S., “Dynamics of premixed confined swirling flames,” Comptes Rendus Mécanique 337 (2009) pp. 395–405.
Pipitone, E. & Mancuso, U., “An experimental investigation of two different methods for swirl induction in a multivalve engine,” International Journal of Engine Research 6 (2005) pp. 159-170.
Reeb, G., “Sur Certaines Propriétés Topologiques des Variétés Feuillétées” (“On Certain Topological Properties of Foliation Varieties”), Actualités Scientifiques et Industrielles, Hermann, Paris (1952).
Roberts, J. B., “Coherence Measurements in an Axisymmetric Wake,” AIAA Journal 11 (1973) pp. 1569-1571.
Roos, F. W. & Willmarth, W. W., “Some Experimental Results on Sphere and Disk Drag,” AIAA Journal 9 (1971) pp. 285-291.
Saha, A. K., “Three-dimensional numerical simulations of the transition of flow past a cube,” Physics of Fluids 16 (2004) pp. 1630-1646.
Shenoy, A. R. & Kleinstreuer, C., “Flow over a thin circular disk at low to moderate Reynolds numbers,” Journal of Fluid Mechanics 605 (2008) pp. 253-262.
Stanton, T. E. & Marshall, D., “Eddy systems behind discs,” Aeronautical Research Committee Reports and Memoranda, No. 1358 (1932).
Stöhr, M., Boxx, I., Carter, C., & Meier, W., “Dynamics of lean blowout of a swirl-stabilized flame in a gas turbine model combustor,” Proceedings of the Combustion Institute 33 (2011) pp. 2953-2960.
Syred N. & Beér, J. M., “Combustion in Swirling Flows: A Review,” Combustion and Flame 23 (1974) pp. 143~201.
Syred, N., “A review of oscillation mechanisms and the role of the precessing vortex core (PVC) in swirl combustion systems,” Progress in Energy and Combustion Science 32 (2006) pp. 93–161.
Tian, X., Ong., M. C., Yang, J., & Myrhaug, D., “Large-eddy simulations of flow normal to a circular disk at Re = 1.5×105,” Computers & Fluids 140 (2016) pp. 422-434.
Turkington, B., “Vortex Rings with Swirl: Axisymmetric Solutions of the Euler Equations with Nonzero Helicity,” SIAM J. Math. Anal. 20 (1989) pp. 57-73.
Willmarth, W. W., Hawk, N. E., & Harvey, R. L., “Steady and unsteady motions and wakes of freely falling disks,” Physics of Fluids 7 (1964) pp. 197-208.
Yang, J., Wub, G., Zhong, W., & Liu, M., “Numerical study on bifurcations in the wake of a circular disk,” International Journal of Computational Fluid Dynamics 28 (2014) pp. 187-203.