Fluid Instability in Concentric rotating cylinders by O3L09H

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									     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


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               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)
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            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

								
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