Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-21T19:59:11.889Z Has data issue: false hasContentIssue false
  • English
  • Français

Dye visualization near a three-dimensional stagnation point: application to the vortex breakdown bubble

Published online by Cambridge University Press:  10 March 2009

M. BRØNS ,
M. C. THOMPSON  and
K. HOURIGAN
M. BRØNS*
Affiliation:
Department of Mathematics, Technical University of Denmark, DK-2800 Lyngby, Denmark
M. C. THOMPSON
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia Division of Biological Engineering, Monash University, Melbourne, Victoria 3800, Australia
K. HOURIGAN
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia Division of Biological Engineering, Monash University, Melbourne, Victoria 3800, Australia
*
†Email address for correspondence: m.brons@mat.dtu.dk
Get access Rights & Permissions [Opens in a new window]

Abstract

An analytical model, based on the Fokker–Planck equation, is constructed of the dye visualization expected near a three-dimensional stagnation point in a swirling fluid flow. The model is found to predict dye traces that oscillate in density and position in the meridional plane in which swirling flows are typically visualized. Predictions based on the model are made for the steady vortex breakdown bubble in a torsionally driven cylinder and compared with computational fluid dynamics predictions and experimental observations. Previous experimental observations using tracer visualization techniques have suggested that even for low-Reynolds-number flows, the steady vortex breakdown bubble in a torsionally driven cylinder is not axisymmetric and has an inflow/outflow asymmetry at its tail. Recent numerical and theoretical studies show that the asymmetry of the vortex breakdown bubble, and consequently its open nature, can be explained by the very small imperfections that are present in any experimental rig. Distinct from this, here it is shown that even for a perfectly axisymmetric flow and breakdown bubble, the combined effect of dye diffusion and the inevitable small errors in the dye injection position lead to the false perception of an open bubble structure with folds near the lower stagnation point. Furthermore, the asymmetries in the predicted flow structures can be remarkably similar to those observed in flow observations and computational predictions with geometric asymmetries of the rig. Thus, when interpreting dye-visualization patterns in steady flow, even if axisymmetric flow can be achieved, it is important to take into account the relative diffusivity of the dye and the accuracy of its injection.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 622 , 10 March 2009 , pp. 177 - 194
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bakker, P. G. 1991 Bifurcations in Flow Patterns. Kluver.CrossRefGoogle Scholar
Brøns, M. 2007 Streamline topology – patterns in fluid flows and their bifurcations. Adv. Appl. Mech. 41, 1–43.CrossRefGoogle Scholar
Brøns, M., Shen, W. Z., Sørensen, J. N. & Zhu, W. J. 2007 The influence of imperfections on the flow structure of steady vortex breakdown bubbles. J. Fluid Mech. 578, 453–466.CrossRefGoogle Scholar
de Bruijn, N. G. 1970 Asymptotic Methods in Analysis. North-Holland.Google Scholar
Chong, M. S., Perry, A. E. & Cantwell, B. J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids A 2 (5), 765–777.CrossRefGoogle Scholar
Cimbala, H. M., Nagib, H. M. & Roshko, A. 1981 Wake instability leading to new large scale structures downstream of bluff bodies. Bull. Am. Phys. Soc. 26, 1256.Google Scholar
Cimbala, H. M., Nagib, H. M. & Roshko, A. 1988 Large structure in the far wakes of two-dimensional bluff bodies. J. Fluid Mech. 190, 265–298.CrossRefGoogle Scholar
Delery, J. M. 1994 Aspects of vortex breakdown. Prog. Aerospace Sci. 30, 1–59.CrossRefGoogle Scholar
Dusting, J., Sheridan, J. & Hourigan, K. 2004 Flows within a cylindrical cell culture bioreactor with a free-surface and a rotating base. In Proc. 15th Australasian Fluid Mechanics Conf. (ed. Behnia, M., Lin, W. & McBain, G. D.). University of Sydney, Sydney, Australia, Paper No. AFMC00155.Google Scholar
Dusting, J., Sheridan, J. & Hourigan, K. 2006 A fluid dynamics approach to bioreactor design for cell and tissue culture. Biotechnol. Bioengng 94, 1197–1208.CrossRefGoogle ScholarPubMed
Fountain, G., Khakhar, D. & Ottino, J. 1998 Visualization of three-dimensional chaos. Science 281, 683–686.CrossRefGoogle ScholarPubMed
Gardiner, C. W. 1985 Handbook of Stochastic Methods, 2nd edn.Springer.Google Scholar
Gelfgat, A. Y. 2002 Three-dimensionality of trajectories of experimental tracers in a steady axisymmetric swirling flow: effect of density mismatch. Theor. Comput. Fluid Dyn. 16, 29–41.CrossRefGoogle Scholar
Gelfgat, A. Y., Bar-Yoseph, P. Z. & Solan, A. 1996 Stability of confined swirling flow with and without vortex breakdown. J. Fluid Mech. 311, 1–36.CrossRefGoogle Scholar
Ghoniem, A. F. & Sherman, F. S. 1985 Grid-free simulation of diffusion using random walk methods. J. Comput. Phys. 61, 1–37.CrossRefGoogle Scholar
Goldshtik, M. & Hussain, F. 1998 Analysis of inviscid vortex breakdown in a semi-infinite pipe. Fluid Dyn. Res. 23, 189–234.CrossRefGoogle Scholar
Gursul, I. & Rockwell, D. 1991 Effect of concentration of vortices on streakline patterns. Exps. Fluids 10, 294–296.CrossRefGoogle Scholar
Gursul, I., Lusseyran, D. & Rockwell, D. 1990 On interpretation of flow visualization of unsteady shear flows. Exps. Fluids 9, 257–266.CrossRefGoogle Scholar
Hama, F. R. 1962 Streaklines in a perturbed shear flow. Phys. Fluids 5, 644–650.CrossRefGoogle Scholar
Hartnack, J. N., Brøns, M. & Spohn, A. 2000 The role of asymmetric perturbations in steady vortex breakdown bubbles. DCAMM Rep. 628. Technical University of Denmark.Google Scholar
Holmes, P. 1984 Some remarks on chaotic particle paths in time-periodic, three-dimensional swirling flows. Contemporary Math. 28, 393–404.CrossRefGoogle Scholar
Hourigan, K., Graham, L. W. & Thompson, M. C. 1995 Spiral streaklines in pre-vortex breakdown regions of axisymmetric swirling flows. Phys. Fluids 7 (12), 3126–3128.CrossRefGoogle Scholar
Kline, S. J. 1969 National committee for fluid mechanics films, film notes for flow visualization. Publication 21607, Encylcopedia Britannica Educational Corporation, Chicago, Illinois.Google Scholar
Krause, E. & Liu, C. H. 1989 Numerical studies of incompressible flow around delta and double-delta wings. Z. Flugwiss. Weltraumforschung 13, 291–301.Google Scholar
Kurosaka, M. & Sundram, P. 1986 Illustrative examples of streaklines in unsteady vortices: interpretational difficulties revisited. Phys. Fluids 29 (10), 3474–3477.CrossRefGoogle Scholar
Leibovich, S. 1978 The structure of vortex breakdown. Annu. Rev. Fluid Mech. 10, 221–246.CrossRefGoogle Scholar
Leibovich, S. 1984 Vortex stability and breakdown: survey and extension. AIAA J. 22, 1192–1206.CrossRefGoogle Scholar
Leibovich, S. 1999 Sensitivity of vortex flows and vortex breakdown to farfield conditions. In AIAA 99-3765, 30th AIAA Fluid Dynamics Conf.CrossRefGoogle Scholar
Liow, K. Y. S., Tan, B. T., Thompson, M. C., Hourigan, K. & Thouas, G. 2007 In-silico characterization of the flow inside a novel bioreactor for cell and tissue culture. In Proc. 16th Australasian Fluid Mechanics Conf. (ed. Jacobs, P., McIntyre, T., Cleary, M., Buttsworth, D., Mee, D., Clements, R., Morgan, R. & Lemckert, C.), pp. 361–367. School of Engineering, The University of Queensland.Google Scholar
Lopez, J. M. & Perry, A. D. 1992 Axisymmetric vortex breakdown. Part 3. Onset of periodic flow and chaotic advection. J. Fluid Mech. 234, 449–471.CrossRefGoogle Scholar
Ottino, J., Leong, C., Rising, H. & Swanson, P. 1988 Morphological structures produced by mixing in chaotic flows. Nature 333, 419–425.CrossRefGoogle Scholar
Risken, H. 1989 The Fokker–Planck Equation. Springer.Google Scholar
Rom-Kedar, V., Leonard, A. & Wiggins, S. 1990 An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–395.CrossRefGoogle Scholar
Rusak, Z. 1996 Axisymmetric swirling flow around a vortex breakdown point. J. Fluid Mech. 323, 79–105.CrossRefGoogle Scholar
Sotiropoulos, F. & Ventikos, Y. 2001 The three-dimensional structure of confined swirling flows with vortex breakdown. J. Fluid Mech. 426, 155–175.CrossRefGoogle Scholar
Sotiropoulos, F., Ventikos, Y. & Lackey, T. C. 2001 Chaotic advection in a three-dimensional stationary vortex-breakdown bubble: Sil'nikov's chaos and the devil's staircase. J. Fluid Mech. 444, 257–297.CrossRefGoogle Scholar
Sotiropoulos, F., Webster, D. R. & Lackey, T. C. 2002 Experiments on Lagrangian transport in steady vortex-breakdown bubbles in a confined swirling flow. J. Fluid Mech. 466, 215–248.CrossRefGoogle Scholar
Spall, R. E., Gatski, T. B. & Ash, R. L. 1990 The structure and dynamics of bubble-type vortex breakdown. Proc. R. Soc. Lond. A 429, 612–637.Google Scholar
Spohn, A., Mory, M. & Hopfinger, E. J. 1998 Experiments on vortex breakdown in a confined flow generated by a rotating disc. J. Fluid Mech. 370, 73–99.CrossRefGoogle Scholar
Tan, B. T., Liow, K., Hourigan, K., Sheridan, J. & Thompson, M. C. 2006 Predicting the flow stresses in a simplified bioreactor. In Proc. Eleventh Asian Congress of Fluid Mechanics (ed. Tso, C. P., Viswanath, P., Fukunishi, Y. & Cui, E.), pp. 220–225. Kuala Lumpur, Malaysia.Google Scholar
Taneda, S. 1956 Experimental investigation of the wakes behind cylinders and plates at low Reynolds numbers. J. Phys. Soc. Japan 11, 302–307.CrossRefGoogle Scholar
Thompson, M. C. & Hourigan, K. 2003 The sensitivity of steady vortex breakdown bubbles in confined cylinder flows to rotating lid misalignment. J. Fluid Mech. 496, 129–138.CrossRefGoogle Scholar
Thouas, G. A., Sheridan, J. & Hourigan, K. 2007 A bioreactor model of mouse tumor progression. J. Biomed. Biotechnol. Art. ID 32754, doi:10.1155/2007/32754.CrossRefGoogle Scholar
Williams, D. R. & Hama, F. R. 1980 Streaklines in a shear layer perturbed by two waves. Phys. Fluids 23 (3), 442–447.CrossRefGoogle Scholar
Yu, P., Lee, T. S., Zeng, Y. & Low, T. H. 2005 a Effect of vortex breakdown on mass transfer in a cell culture bioreactor. Mod. Phys. Lett. B 19, 1543–1546.CrossRefGoogle Scholar
Yu, P., Lee, T. S., Zeng, Y. & Low, T. H. 2005 b Fluid dynamics of a micro-bioreactor for tissue engineering. Fluid Dyn. Mat. Process. 1, 235–246.Google Scholar
Yu, P., Lee, T. S., Zeng, Y. & Low, T. H. 2007 Characterization of flow behaviour in an enclosed cylinder with a partially rotating end wall. Phys. Fluids 19, 057104.CrossRefGoogle Scholar
Zienkiewicz, O. C. 1977 The Finite Element Method, 3rd edn.McGraw-Hill.Google Scholar