"Fluid Instability in Concentric rotating cylinders"
Fluid Instability in Concentric Rotating Cylinders Sarah Macumber CS Senior Thesis Advisors: Mike Sprague, Patrick Weidman and Henry Tufo Overview • History • Research Goals • Governing Equations • Geometric Model • Simulation Code • Results • Applications 10/4/2012 Taylor Vortex Flow Taylor’s Experiment • 1923 • Annulus configuration • First case of a stability calculation quantitatively matching experimental values • Now commonly used for quantitative comparison between theory and experiment "The Taylor-Couette system of shear flow in concentric cylinders is a canonical system that provides valuable insight into centrifugal stability of rotating flows as well as low dimension bifurcation phenomena” ~Czarny 10/4/2012 Taylor Vortex Flow Taylor’s Experiment Rotation of the interior cylinder U • U < Uc – Stable circular flow • Circular Couette Flow (CCF) • U > Uc – Flow is unstable to axisymmetric perturbations which form toroidal vortices • Taylor-vortex flow (TVF) • U >> Uc – Flow is unstable to non-axisymmetric perturbations producing azimuthal waves • Wavy Taylor vortex flow (WTV) 10/4/2012 Taylor Vortex Flow 1st and 2nd instabilities TVF Wavy TVF 10/4/2012 Taylor Vortex Flow Senior Thesis • 1st Semester Goals – Governing equations – Non-dimensional parameters – Linear stability analysis – Code Validation using regular geometry 10/4/2012 Taylor Vortex Flow Governing Equations • 3D Navier Stokes – Conservation of Mass – Momentum – Energy • Cylindrical Coordinates • Axisymmetric ∂U / ∂ = 0 • Non-dimensionalization 10/4/2012 Taylor Vortex Flow Governing Equations Non-dimensionalized axisymmetric Navier Stokes for incompressible fluid flow in Cylidrical Coordinates 10/4/2012 Taylor Vortex Flow Non-dimensional parameters • Reynolds number Re – Known for a certain – Re = (L U) / • Ratio of radii – = r1 / r2 L length scale kinematic viscosity U angular velocity profile 10/4/2012 Taylor Vortex Flow Geometry 10/4/2012 Taylor Vortex Flow Geometry • How can we use 2D geometry to model a 3D problem? – Use the energy equation to model the radial component of velocity – (x,y) (z,r) and Temperature = 10/4/2012 Taylor Vortex Flow Simplified 2D Geometry • Does not allow non-axisymmetric flows – No wavy vortices • Concerned only with first onset of instability – CCF TVF • Extremely fast 10/4/2012 Taylor Vortex Flow Boundary and Initial Conditions • For purposes of Code Validation – Ends are periodic: infinitely long cylinder – Interior cylinder rotating at constant – Exterior cylinder is fixed =0 – Initial radial velocity profile of CCF is set from linear stability theory 10/4/2012 Taylor Vortex Flow Numerical method • NEKTON – Henry Tufo of CS – Parallel code (Fortran) • hemisphere cluster – Integration of Navier Stokes equations done using Spectral Element Method – 3 part process • PRENEK • NEK500 • POSTNEK 10/4/2012 Taylor Vortex Flow Velocity in (temperature) 10/4/2012 Taylor Vortex Flow Velocity in Z (x) 10/4/2012 Taylor Vortex Flow Other Experimental Vortices QuickTime™ and a TIFF (LZW) decompressor are neede d to see this picture. Wavy Taylor vortices, Braided Taylor vortices and Turbulent Taylor vortices 10/4/2012 Taylor Vortex Flow What’s left to do? • 2nd Semester Goals – Implement code for irregular geometry – Using simulations study • Bi-modal TVF • Endwall effects (Ekman pumping) • Flow at discontinuities of interior cylinder – Find ultimate configuration for eventual physical realization (DLC) 10/4/2012 Taylor Vortex Flow Theoretical Applications • Taylor Annulus is canonical apparatus for instability analysis – Hydrodynamics – Hydromagnetic flows – Heated flows – Any CFD with rotation • Rotating lid, stationary cylinder 10/4/2012 Taylor Vortex Flow Mechanical Applications • Propulsion • Chemical Mixing • Cooling • Rotor Stator model • Engines • Filtration 10/4/2012 Taylor Vortex Flow Fluid Mixing 10/4/2012 Taylor Vortex Flow Pumping 10/4/2012 Taylor Vortex Flow Filtration • Interior cylinder is porous 10/4/2012 Taylor Vortex Flow Questions? 10/4/2012 Taylor Vortex Flow