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					Simulation of the Initial 3-D
Instability of a Vortex Ring


        Justin Wiwchar
                    Outline
•   Objectives
•   Computational Details
•   Simulation Details
•   Results
•   Conclusions
                  Objectives
• Peck & Sigurdson (1994) observed an
  azimuthal instability in the impacting water
  drop experiment
• Simulate multiple vorticity fields to recreate
  the azimuthal instability
• If a certain vorticity field results in the
  instability, it may also be the source of the
  experimental instability
       Computational Details
• Details of the Code
• Code Verification
          Details of the Code
• Developed by J.H. Walther & P.
  Koumoutsakos
• Simulations performed remotely on a
  supercomputer in Manno, Switzerland
• 3-D viscous vortex method (Lagrangian)
                 Details of the Code
• Navier-Stokes for constant density &
  viscosity:
  Dω
      (ω  )u  ν 2ω  
  Dt


 Tilting & Stretching               Body Forces
                        Diffusion
               Details of the Code
• First step is inviscid, corresponds to the tilting
  & stretching term

        u xi 
   dxi                      Location changed to account for advection
   dt

   dαi
        voli ω xi    u xi                Strength changed to account
                                                  for tilting & stretching
   dt
            Details of the Code
• Second step is viscous, allows for diffusion
  and body forces

  dx i
       0             Location is fixed
  dt

  dαi
  dt
             
       voli  2 ω xi    xi       Strength changes from
                                          diffusion and body
                                          forces
            Code Verification
• Thin-cored vortex ring was simulated,
  convection speed was compared to theoretical
  values
• Convection speed of the centroid of a thin-
  cored vortex ring with an Oseen core:

               8R
  Uc       (ln       0.558 )
       4 R     4 t
                                                 Code Verification
                                       Comparison of Theoretical and Simulated Convection Speeds For a Thin-Cored Vortex Ring

                             5.0


                             4.5


                             4.0


                             3.5
Convection Speed, |4πRU/Γ|




                             3.0


                             2.5


                             2.0


                             1.5

                                                                                                          Theoretical
                             1.0
                                                                                                          Simulated (a/R=0.05)
                                                                                                          Simulated (a/R=0.1)
                             0.5
                                                                                                          Simulated (a/R=0.2)

                             0.0
                             0.0E+00   1.0E-04    2.0E-04     3.0E-04     4.0E-04          5.0E-04   6.0E-04       7.0E-04       8.0E-04
                                                                                       2
                                                                        Time, νt/16R
            Code Verification
• At a core radius ratio of α/R = 0.05 the max
  error was -9.4%
• For most of the simulation it was closer to -3%
• Simulation may not have been suitably thin-
  cored
• Core is allowed to diffuse so simulation and
  theory will disagree at later times
           Simulation Details
• Instability Hypothesis
  – Solitary ring
  – Opposite-signed ring
  – Vortex breakdown
• Details of Perturbations Used
  – Random perturbation
  – Wave-number perturbation
               Solitary Ring
• Single vortex ring with no other outside
  vorticity
• Could trigger a Widnall instability
         Opposite-Signed Ring
• Smaller, weaker secondary vortex ring placed
  above the primary ring
• Could trigger a Rayleigh centrifugal instability
           Vortex Breakdown
• Image vortex ring placed above the primary
  ring
• Could trigger a vortex breakdown instability
              Random Perturbation
• Every point of vorticity in the domain had its
  magnitude altered based on a random number
• Perturbation was initially based on the global
  vorticity
• Later changed the perturbation to be based on
  the local vorticity (areas of high vorticity
  receive higher perturbations)
  ωperturbed = ωinitial + 2 * (Max perturbation) * (Random number - 0.5)
Random Perturbation
        Wave-number Perturbation
• 30 wave-numbers were added to the ring
• Each had a random phase and equal amplitude
• Amplitude of each wave-number was 1% r0 for
  the majority of the simulations

  do i = 1,30
  x = x + amplitude * sin (i * θ + 2 * π * phasei) * cos (θ)
  y = y + amplitude * sin (i * θ + 2 * π * phasei) * sin (θ)
  end do
Wave-number Perturbation
                     Results
•   Solitary ring
•   Opposite-signed ring
•   Vortex breakdown
•   Skeleton vortex structure
•   Instability growth
•   Source of the instability
Solitary Ring
Solitary Ring
            Solitary Ring




Re = 1400              Re = 2500
Opposite-signed Ring
Opposite-signed Ring
Vortex Breakdown
      Skeleton Vortex Structure
• Vortex lines were calculated through
  multiple points, chosen from a contour plot
  on an x-y plane through the centroid of the
  ring
• The vortex lines were calculated at
  multiple time-steps to observe how they
  evolved
Skeleton Vortex Structure
     Skeleton Vortex Structure
              Vortex Line 1a




t = 0.001 s    t = 0.005 s     t = 0.009 s
     Skeleton Vortex Structure
              Vortex Line 1b




t = 0.001 s    t = 0.005 s     t = 0.009 s
     Skeleton Vortex Structure
              Vortex Line 2




t = 0.001 s    t = 0.005 s    t = 0.009 s
            Instability Growth
• Slices were taken in the x-y plane through the
  centroid of the vortex ring
• ωr and ωz were calculated azimuthally around
  the core of the vortex ring
• A Fourier transform was performed on the ωr
  and ωz data to determine the dominant wave-
  numbers
• The analysis was performed at each time-step
  to examine the growth of each wave-number
Instability Growth
Instability Growth
Instability Growth
          Source of the Instability
•   Widnall instability
•   Rayleigh centrifugal instability
•   Vortex breakdown
•   Elliptic instability
           Widnall Instability
• Simulations run with the solitary ring did
  develop an azimuthal instability
• The signature “waviness” of the core which
  would be expected from a Widnall instability
  is not visible
• Likely not a Widnall instability
  Rayleigh Centrifugal Instability
• Simulations run with the opposite-signed
  secondary vortex ring also produced an
  azimuthal instability
• Appearance of the instability does not
  significantly change from what is seen with the
  solitary ring
• Likely not the Rayleigh centrifugal instability,
  however, the presence of the opposite-signed
  vorticity may amplify the instability
           Vortex Breakdown
• Simulations run with an image vortex
  produced the azimuthal instability
• Again, appearance didn’t change noticeably
  from simulations with the solitary ring
• Possible that the ring diameter contraction was
  not strong enough to set off a vortex
  breakdown instability
            Elliptic Instability
• Leweke & Williamson (1998) observed elliptic
  streamlines in counter-rotating vortex pairs
• Slices through the x-z or y-z planes reveal that
  the core goes elliptical in the solitary ring
  simulations
• Ring also develops a “slant” where the two
  sides of the core are displaced in the z-
  direction
• Likely an elliptic instability
                 Conclusions
• Instabilities were only observed when the
  wave-number perturbation was applied
• Instabilities were observed with all three
  vorticity fields: the solitary ring, the opposite-
  signed ring, and the vortex breakdown
  simulations
• Wave-number 1 is dominant when examining
  either ωr or ωz.
                Conclusions
• Higher wave-numbers are insignificant at Re =
  1400 but significant at Re = 2500
• Examining vortex lines gives an idea how the
  instability develops
• The elliptic instability is likely the source of
  the bracelet structure seen experimentally

				
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