Mixing Through Stirring of Steady Flow in Small Amplitude Helical Tubes
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Abstract
In this paper we numerically simulate flow in a helical tube for physiological conditions using a co-ordinate mapping of the Navier–Stokes equations. Helical geometries have been proposed for use as bypass grafts, arterial stents and as an idealized model for the out-of-plane curvature of arteries. Small amplitude helical tubes are also currently being investigated for possible application as A–V shunts, where preliminary in vivo tests suggest a possibly lower risk of thrombotic occlusion. In-plane mixing induced by the geometry is hypothesized to be an important mechanism. In this work, we focus mainly on a Reynolds number of 250 and investigate both the flow structure and the in-plane mixing in helical geometries with fixed pitch of 6 tube diameters (D), and centerline helical radius ranging from 0.1D to 0.5D. High-order particle tracking, and an information entropy measure is used to analyze the in-plane mixing. A combination of translational and rotational reference frames are shown to explain the apparent discrepancy between flow field and particle trajectories, whereby particle paths display a pattern characteristic of a double vortex, though the flow field reveals only a single dominant vortex. A radius of 0.25D is found to provide the best trade-off between mixing and pressure loss, with little increase in mixing above R = 0.25D, whereas pressure continues to increase linearly.
Keywords
Laminar Spectral/hp Graft Shunt Stent Pipe Thrombosis Co-ordinate mapping AdvectionNotes
Acknowledgment
The authors would like to acknowledge the EPSRC for funding this research.
References
- 1.Bryan, A. J., and G. D. Angelini. The biology of saphenous vein graft occlusion: etiology and strategies for prevention. Current opinion in cardiology 9(6), 1994.PubMedCrossRefGoogle Scholar
- 2.Butler, J. P., Tsuda, A., 1997. Effect of convective stretching and folding on aerosol mixing deep in the lung, assessed by approximate entropy. J Appl Physiol 83, 800–809.PubMedGoogle Scholar
- 3.Caro, C. G., Cheshire, N. J., Watkins, N., 2005. Preliminary comparative study of small amplitude helical and conventional eptfe arteriovenous shunts in pigs. Journal of the Royal Society interface 2, 261–266.PubMedCrossRefGoogle Scholar
- 4.Caro, C. G., Doorly, D. J., Tarnawski, M., Scott, K. T., Long, Q., Dumoulin, C. L., 1996. Non-planar curvature and branching of arteries and non-planar-type flow. Proceedings: Mathematical, Physical and Engineering Sciences 452 (1944), 185–197.CrossRefGoogle Scholar
- 5.Coppola, G., Caro, C., 2008. Oxygen mass transfer in a model three-dimensional artery. Journal of the Royal Society Interface 5 (26), 1067–1075.PubMedCrossRefGoogle Scholar
- 6.Coppola, G., S. J. Sherwin, and J. Peiro. Non-linear particle tracking for high-order elements. J. Comput. Phys. 172, 2001.Google Scholar
- 7.Darekar, R. M., Sherwin, S. J., 2001. Flow past a square-section cylinder with a wavy stagnation face. Journal of Fluid Mechanics 426, 263–295.CrossRefGoogle Scholar
- 8.Doorly, D. J., 1999. Flow transport in arteries. In: Sajjadi, S. G., Nash, G. B., Rampling, M. W. (Eds.), Cardiovascular Flow Modelling And Measurement With Application To Clinical Medicine. Oxford University Press, pp. 67–81.Google Scholar
- 9.Doorly, D. J., Sherwin, S. J., Franke, P. T., Peiro, J., 2002. Vortical flow structure identification and flow transport in arteries. Computer Methods in Biomechanics and Biomechanical Engineering 5 (3), 261–275.CrossRefGoogle Scholar
- 10.Evangelinos, C. Parallel spectral/hp methods and simulations of flow/structure interactions. Ph.D. thesis, Brown University, 1999.Google Scholar
- 11.Friedman, M. H., 1993. Arteriosclerosis research using vascular flow models: from 2-d branches to compliant replicas. Journal of biomechanical engineering 115 (4B), 595–601.PubMedCrossRefGoogle Scholar
- 12.Germano, M., 1981. On the effect of torsion on a helical pipe flow. Journal of Fluid Mechanics 125, 1–8.CrossRefGoogle Scholar
- 13.Huijbregts, H. J. T. A. M., Blankestijn, P. J., Caro, C. G., Cheshire, N. J. W., Hoedt, M. T. C., Tutein~Nolthenius, R. P., Moll, F. L., 2007. A helical ptfe arteriovenous access graft to swirl flow across the distal anastomosis: Results of a preliminary clinical study. European Journal of Vascular and Endovascular Surgery 33 (4), 472–475.PubMedCrossRefGoogle Scholar
- 14.Kang, T. G., Kwon, T. H., 2004. Colored particle tracking method for mixing analysis of chaotic micromixers. Journal of Micromechanics and Microengineering 14, 891.CrossRefGoogle Scholar
- 15.Khakhar, D. V. Analysis of chaotic mixing in two model systems. J. Fluid Mech. 172, 1986.CrossRefGoogle Scholar
- 16.Koberg, W. H. Turbulence control for drag reduction with active wall deformation. Ph.D. thesis, Imperial College London, 2008.Google Scholar
- 17.Krasnopolskaya, T. S., Meleshko, V. V., Peters, G. W. M., Meijer, H. E. H., 1999. Mixing in stokes flow in an annular wedge cavity. European journal of mechanics. B, Fluids 18 (5).CrossRefGoogle Scholar
- 18.Leuprecht, A., Perktold, K., Prosi, M., Berk, T., Trubel, W., Schima, H., 2002. Numerical study of hemodynamics and wall mechanics in distal end-to-side anastomoses of bypass grafts. Journal of Biomechanics 35, 225–236.PubMedCrossRefGoogle Scholar
- 19.Loth, F., Fischer, P. F., Bassiouny, H.-S., 2008. Blood flow in end-to-side anastomoses. Annual Review of Fluid Mechanics 40, 367–393.CrossRefGoogle Scholar
- 20.Newman, D. A computational study of fluid/structure interactions: flow-induced vibrations of a flexible cable. Ph.D. thesis, Princeton University, 1996.Google Scholar
- 21.Nielsen, L. B., 1996. Transfer of low density lipoprotein into the arterial wall and risk of atherosclerosis;. Atherosclerosis 123 (1–2), 1–15.PubMedCrossRefGoogle Scholar
- 22.Ramstack, J. M., Zuckerman, L., Mockros, L. F., 1979. Shear-induced activation of platelets. Journal of Biomechanics 12 (2), 113–25.PubMedCrossRefGoogle Scholar
- 23.Shannon, C. E., 1948. A mathematical theory of communication. Bell System Technical Journal 27, 379–423.Google Scholar
- 24.Sherwin, S. J., Karnaidakis, G. E., 2005. Spectral/hp Element Methods for Computational Fluid Dynamics. Oxford Science Publications.Google Scholar
- 25.Sherwin, S. J., Shah, O., Doorly, D. J., Peiro, J., Papaharilaou, Y., Watkins, N., Caro, C. G., Dumoulin, C. L., 2000. The influence of out-of-plane geometry on the flow within a distal end-to-side anastomsis. ASME J. Biomech 122, 86–95.CrossRefGoogle Scholar
- 26.Wang, C. Y., 1981. On the low-reynolds-number flow in a helical pipe. Journal of Fluid Mechanics 108, 185–194.CrossRefGoogle Scholar
- 27.Wang, W., Manas-Zloczower, I., Kaufman, M., 2003. Entropic characterization of distributive mixing in polymer processing equipment. AIChE Journal 49, 1637–1644.CrossRefGoogle Scholar
- 28.Wang, W., Manas-Zloczower, I., Kaufman, M., 2005a. Entropy time evolution in a twin-flight single-screw extruder and its relationship to chaos. Chemical Engineering Communications 192, 405–423.CrossRefGoogle Scholar
- 29.Wang, W., Manas-Zloczower, I., Kaufman, M., 2005b. Influence of initial conditions on distributive mixing in a twin-flight single-screw extruder. Chemical Engineering Communications 192, 749–757.CrossRefGoogle Scholar
- 30.Yamamoto, K., Aribowo, A., Hayamizu, Y., Hirose, T., Kawahara, K., 2002. Visualization of the flow in a helical pipe. Fluid Dynamics Research 30, 251–267.CrossRefGoogle Scholar
- 31.Yamamoto, K., Yanase, S., Jiang, R., 1998. Stability of the flow in a helical tube. Fluid Dynamics Research 22, 153–170.CrossRefGoogle Scholar
- 32.Zabielski, L., Mestel, A. J., 1998. Steady flow in a helically symmetric pipe. Journal of Fluid Mechanics 370, 297–320.CrossRefGoogle Scholar