Vortex detection in time-dependent
flow
Ronny Peikert
ETH Zurich
Vortex detection - early work
• Derived from physical properties:
– vortex regions:
• Pressure Laplacian
• Q criterion (Okubo-Weiss, Hunt 1991)
• l2 criterion (Jeong and Hussain, 1995)
– vortex axes (vortex core lines):
• Pressure&vorticity based (Banks and Singer, 1994)
• Pressure valley line (Kida and Miura, 1997)
These are valid for steady and unsteady flow!
Dagstuhl Seminar Scientific Visualization, July 15-20, 2007
2
Vortex detection - early work (2)
• Derived from geometric/topological properties:
– vortex axes:
• critical point analysis, separatrices (Helman and
Hesselink 1991, Globus et al. 1991)
• helicity-based, Levy et al. (1990)
• streamline-based, Sujudi and Haimes (1995)
• higher-order, Roth and Peikert (1998)
These are just formulated for steady flow!
But vortex axes are useful, complementary to vortex
regions!
Dagstuhl Seminar Scientific Visualization, July 15-20, 2007
3
Vortex detection - more recent work
• Lagrangian type methods:
– Non-local swirl [Cucitore 1999]
– Objective criterion Mz [Haller 2005]
Better than l2 ?
• Time-dependent vortex axes methods:
– Swirling particle motion [Weinkauf et al. 2007]
– Work in progress [Bürger et al.]
Dagstuhl Seminar Scientific Visualization, July 15-20, 2007
4
Adaptations of Sujudi-Haimes criterion
• Sujudi-Haimes criterion (in parallel vectors formulation)
εr  realEigenvector u
εr u AND filter criteria
• Equivalent and more efficiently computable:
as u AND filter criteria
as  u u
• Time-dependent version:
at u AND filter criteria at   u  u  u  acceleration
t
• Weinkauf et al. (2007) (equivalent formulation):
at εr AND filter criteria
Dagstuhl Seminar Scientific Visualization, July 15-20, 2007
5
Tilting vortex example
  y  tz 


u  x, y , z, t    x  tz 
 z 


Sujudi-Haimes axis  tz,0, z 
streamlines
at t=0.3
Dagstuhl Seminar Scientific Visualization, July 15-20, 2007
tz,0, z 
tz, tz, z 
pathlines
seeded
at t=0.3
6
Vortex rope example
Dagstuhl Seminar Scientific Visualization, July 15-20, 2007
7
Synthetic vortex rope
streamlines
pathlines
Dagstuhl Seminar Scientific Visualization, July 15-20, 2007
8
Radii of vortex core lines
Unsteady flow
Levy et al.
k 

1

 2s  R


Sujudi / Haimes
 k
1 s  R


 k  
1 s  R


 k  k 2 
1      R

s  s  

1
k


1

 s R


 k    k   2 

1
R


s
 s  

Higher-order
Correct
Dagstuhl Seminar Scientific Visualization, July 15-20, 2007
9
1
 k  
1 s  R


adapted methods
Steady flow
Comparisons
Comparison of at u and at εr :
• Both reduce to Sujudi-Haimes criterion if flow is steady.
• Criterion at εr is Galilean invariant.
• Consequently, it produces Sujudi-Haimes vortex core
lines also in linearly moving frame of reference.
• Visually indistinguishable in synthetic vortex rope and
CFD vortex rope examples.
• Criterion at u is possibly better in other CFD dataset
examples.
Dagstuhl Seminar Scientific Visualization, July 15-20, 2007
10
Comparisons (2)
• Comparison (Tufo et al. '99)
–
–
–
–
mean velocity
rms velocity
pressure
spanwise vorticity
– l2
• Vortex core line methods
have problems with mixing
layer vortices
• How about Mz ?
Dagstuhl Seminar Scientific Visualization, July 15-20, 2007
11
Conclusion
• Existing methods need further comparison
• What degree of invariance is needed?
Galilean? Objective?
• We need a topology of time-dependent vector
fields
Dagstuhl Seminar Scientific Visualization, July 15-20, 2007
12