Visualization Tools for
Vorticity Transport Analysis
in Incompressible Flow
November 2006 - IEEE Vis
Filip Sadlo, Ronald Peikert @ CGL - ETH
Zurich
Mirjam Sick @ VA TECH HYDRO Switzerland
Motivation
• Analyze vortex creation/dynamics
Vortex core lines (black)
Vorticity Transport Analysis ...
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Motivation
• Analyze vortex creation/dynamics
Vortex core lines (black)
Vorticity Transport Analysis ...
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Motivation
• Analyze vortex creation/dynamics
Vortex core lines (black)
Vorticity Transport Analysis ...
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Motivation
• Analyze vortex creation/dynamics
Vortex core lines (black)
Vorticity Transport Analysis ...
Upstream path lines
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Motivation
• Vortices and shear flow closely related
 Analysis of vorticity w (curl of velocity:
u)
• Vortex lines only frozen in ideal fluids (n =
0)
 Vorticity Transport Analysis
– Based on vorticity equation:
Dw/Dt = … (see later)
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Motivation
• Avoid integration of quantities along paths
–
–
–
–
Accumulation of error
Too high simulation error in practical CFD
Additional parameters
Expensive
• Quantities locally in space-time
– Advection aspect by pathlines + derivatives
– Static visualization
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Overview
•
•
•
•
•
•
Related Work
Vorticity Equation
Quantities for Visualization
Visualization Methods
Applications
Conclusion
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Related Work
• Vortex core lines
–
–
–
–
Levy et al. 1990: based on helicity (uw)
Banks et al. 1995: w-predictor, p-corrector
Strawn et al. 1998: height ridges of ||w||
Sahner et al. 2005: valley lines of l2
• Vortex regions
– Jeong et al. 1995: l2: based on eigenvalues of S2 + W2 of
u
– Silver et al. 1996: tracking of isosurfaces of ||w||
• Vortex lines
– Sadlo et al. 2004: vortex lines with ||w||-proportional
density
• Stream surface based
– Laramee et al. 2006: w-texture advection on stream
surfaces
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Vorticity Equation
• Navier-Stokes
u
p
 u u  
 n 2u
t

 Vorticity Equation
ω
 u ω  ω u  n 2ω
t
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• velocity u, pressure p
• uniform density 
• uniform viscosity n
Quantities for Visualization
• Vorticity Equation
ω
 u  ω  ω  u  n 2ω
t
stretching/tilting
ω
stretching
ω
ω
tilting
ω
diffusion
 ω u  ω ω u ω
• Restrict analysis to ||w||
 ω

2

u


ω

   ω  u  ω  n ω  ω    
 t
ω
      LHS ( 0 because of numerics)
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Vorticity Equation and
Turbulence Models
• Two-equation turbulence models (k-e, k-w,
SST)
 Introduce modified pressure, modified viscosity
• Navier-Stokes
u
p
t
 u u  

n e 2u  2S n e
ωVorticity
Equation
 u ω  ω u  n  2ω  n
t
e
• velocity u, pressure p’
• uniform density 
• non-uniform viscosity ne
• S   u   u T  / 2
2


u     2S n e 
e
additional diffusion terms
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Quantities for Visualization
for Non-Uniform Viscosity
• Vorticity Equation
ω
 u ω  ω u  n e 2ω  n e   2u     2S n e 
t
stretching/tilting
diffusion
 ω u  ω ω u ω
• Again, restrict analysis to ||w||
 ω

2
2

u

ω

ω

u

n

ω


n

u    2S n e      



e
e


ω
ω
 t
ω
      LHS
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( 0 because of numerics)
Visualization Methods:
Pathline Plots
>0
<0
>0
<0

|| w ||
• pathline (fits D/Dt)
• plot ||w|| along pathline
• , : bands around ||w||
• pos. above, neg. below
• ,  decompose D||w||/Dt
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Visualization Methods:
Striped Pathlines
• tube around pathline
• tube radius: ||w||
• color code for each segment
 data stripes
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Visualization Methods:
Striped Pathlines
• tube around pathline
• tube radius: ||w||
• color code for each segment
 data stripes + error stripes
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Visualization Methods:
Striped Pathlines
(a) Evenly-timed segments (show velocity)
(b) Evenly-spaced segment lengths
(c) With error stripes
(d) Normalized data stripes
(e) Scaling instead of normalization
(f) As (a) with striped slices
(g) With error stripes
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Applications:
Separation Vortex
vorticity streamlets
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Applications:
Separation Vortex
vortex (high helicity)
shear flow (low helicity)
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Applications:
Separation Vortex
gain by stretching and
loss by diffusion
almost pure advection
diffusion from boundary
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Applications:
Separation Vortex
Linked view
wall distance
indicators
boundary shear flow
(low wall distance)
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Applications:
Recirculation and Vortex
vortex
recirculation
zone
boundary shear flow
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Applications:
Recirculation and Vortex
gain by stretching
loss by diffusion
loss by stretching
and diffusion
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Applications:
Bifurcation
reception of vorticity
from boundary shear
gain by stretching
loss by diffusion
almost pure
advection
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Applications:
Bifurcation
Courant number indicating
high simulation error
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Applications:
Transient Vortex Rope
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Applications:
Transient Vortex Rope
diffusion front of
boundary shear flow
frequencies of wall distance and
stretching sign differ -> alternating
sign due to moving vortex
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Conclusion
• Tools for vorticity transport analysis
– Allow analysis of vortex dynamics
• Results well consistent with theory
– Vorticity cannot be created inside fluid with
constant density (baroclinic vorticity
generation)
 Usually advected from shear flow at the
boundaries
– Dominant mechanism in vortex regions: gain
by vortex stretching together with loss by
diffusion
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End
Thank you for your attention.
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