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Taylor Couette Flow
TaylorCouette flow is the flow of a viscous fluid sheared in the gap between two rotating coaxial cylinders. Sometimes called the “hydrogen atom of hydrodynamics,” TaylorCouette flow has a rich phenomenology and has long been a testbed for hydrodynamic stability theory and an important tool in the study of nonlinear dynamics and pattern formation theory. Because of its simplicity, the TaylorCouette system is an ideal platform for teaching students about these important physics topics, but also about the fabrication of scientific apparatus. Figure 1. Basic TaylorCouette cell. The basic apparatus consists of two coaxial cylinders. The gap between them is filled with fluid and the inner cylinder is made to rotate by a stepper motor (see Fig. 1). For low rotation rates, viscosity causes the fluid to flow as might be predicted from symmetry. Fluid elements follow circular paths and the flow field has no axial or azimuthal dependence. However, as the rotation rate of the inner cylinder is increased the flow suddenly undergoes an instability and a cellular pattern of toroidal vortices called Taylor rolls emerges. As the rotation is increased further, the Taylor rolls themselves become unstable giving way to a progressively more complicated flow states as shown in Fig. 2, eventually leading to turbulence. Figure 2. As the rotation rate of the inner cylinder is increased (left to right), the flow undergoes bifurcations from Taylor rolls to progressively more complex and eventually turbulent flows. During the first day of the Immersion participants will learn how to build a small TaylorCouette apparatus and how the design and construction of the apparatus can be implemented as a series of instructional modules on digital electronics, computeraided design (CAD), conventional and modern fabrication methods (soldering, machining, laser cutting, 3D printing, etc. as appropriate to participant’s home institutions).
During the second part of the Immersion (2 days), participants will learn how the TaylorCouette system can be used to demonstrate a host of important topics in fluid mechanics and nonlinear dynamics as standalone classroom demonstrations, laboratory exercises, or extended student projects. Some of the topics that will be explored include flow visualization, Reynolds similarity, flow instabilities, the time reversibility of low Reynolds number flows, the coexistence of attractors in nonlinear systems, the RuelleTakens route to chaos, and attractor reconstruction from time series. For the 2019 ALPhA/PICUP Immersion at the University of WisconsinRiver Falls, special attention will be paid to computational analysis of digital images, including spectral and time series analysis. We will also cover numerical modeling of the advection of passive tracers in circular Couette flow using GlowScript and numerical modeling of the primary instability using Daedalus, a Pythonbased package for computational fluid dynamics.
Participants should bring their lab notebook and a flash drive.
There are no special safety considerations besides the usual hazards involved in working with hand and machine tools. The apparatus can be constructed as student projects for ~$200 per unit. Most of the parts can be reused, so the design and construction of the apparatus can be implemented as a projectbased activity in subsequent semesters at significantly lower cost. While the TaylorCouette system can be used to perform most of the experiments as a standalone unit, additional equipment ranging in cost from $200 to $2000 is necessary to conduct some of the experiments.
Fabrication Notes:
CAD Drawings:
 OnShape models of parts for laser cutting/conventional machining. CAD models of commercial parts from McMasterCarr.
Parts List:
 List of parts with vendors, model numbers, links, and pricing as of 4/19/2019.
Computational Activities:
Dye Advection in Circular Couette Flow:
Simulation of Axisymmetric TaylorCouette Flow:
 Direct numerical simulation of incompressible, axisymmetric TaylorCouette flow using Dedalus.
Relevant Literature:
General References:
 R. Tagg, "The CouetteTaylor Problem," Nonlinear Science Today 4, 125 (1994).
 E.L. Koschmieder, Bénard Cells and Taylor Vortices, (Cambridge University Press, Cambridge UK, 1993).
 R. Tagg, "TaylorCouette Reference List." University of ColoradoDenver (1997).
Flow Visualization:
 D. BorreroEcheverry, C.J. Crowley, and T.P. Riddick, "Rheoscopic Fluids for a PostKalliroscope World," Phys. Fluids 30, 087103 (2018).
 E. Fonda and K.R. Sreenivasan, "Unmixing demonstration with a twist: A photochromic TaylorCouette device," Am. J. Phys. 85, 796 (2017).
 P. D. Weidman, “Measurement Techniques in Laboratory Rotating Flows” in Advances in Fluid Mechanics Measurements (SpringerVerlag, Heidelberg) 412–418 (1989).
Low Reynolds Number Flows:
 R. Fitzpatrick, "TaylorCouette Flow," in Fluid Mechanics (2016).
 G.I. Taylor, "Film Notes for Low Reynolds Number Flow," in Illustrated Experiments in Fluid Mechanics: The National Committee for Fluid Mechanics Films Book of Film Notes (Educational Services Inc., Chicago, 1967). video
 E.M. Purcell, "Life at low Reynolds Number," Am. J. Phys. 45, 311 (1977).
 S. Childress, "Chapter 7: Stokes Flow" in An Introduction to Theoretical Fluid Mechanics (American Mathematical Society/Courant Institute of Mathematical Sciences, New York, 2009).
Primary Instability:
 G.I. Taylor, "VIII. Stability of a Viscous Liquid Contained Between Two Rotating Cylinders," Phil. Trans. R. Soc. London, Ser. A 223, 289343 (1923).
 C.C. Lin, The Theory of Hydrodynamic Stability, (Cambridge University Press, Cambridge, 1955).
 P.G. Drazin and W.H. Reid, Hydrodynamic Stability 2^{nd} ed., (Cambridge University Press, Cambridge, 2004).
Transition to Turbulence:
 D. Coles, "Transition in Circular Couette Flow," J. Fluid Mech. 21, 385425 (1965).
 J.P. Gollub and H.L. Swinney, "Onset to Turbulence in a Rotating Fluid," Phys. Rev. Lett. 35, 927 (1975).
 C.D. Andereck, S.S. Liu, and H.L. Swinney, "Flow Regimes in a Circular Couette System with Independently Rotating Cylinders," J. Fluid Mech. 164, 155183 (1986).
Attractor Reconstruction:
 N.H. Packard, J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, "Geometry from a Time Series," Phys. Rev. Lett. 45, 712 (1980).
 A. Brandstäter, J. Swift, H.L. Swinney, A. Wolf, J.D. Farmer, E. Jen, and P.J. Crutchfield, "LowDimensional Chaos in a Hydrodynamic System," Phys. Rev. Lett. 51, 1442 (1983).
 A.M Fraser and H.L. Swinney, "Independent coordinates for strange attractors from mutual information," Phys. Rev. A. 33, 1134 (1986).
