Tutorial: Visualization of Time-Varying Vector Fields Part II: Lagrangian Visualization Filip Sadlo VISUS Universität Stuttgart Germany
Overview Part I: Vortices Part II: Vector Field Topology
Part I: Vortices
Vortex Definition No rigorous, widely-accepted definition Robinson 1991: A vortex exists when instantaneous streamlines mapped onto a plane normal to the vortex core exhibit a roughly circular or spiral pattern, when viewed from a reference frame moving with the center of the vortex core Requires a priori detection Not Galilean invariant
Vortex Criteria: Δ Δ > 0 [Chong et al. 1990] Eigenvalues λ 1, λ 2, λ 3 of u satisfy characteristic equation λ 3 Pλ 2 + Qλ R = 0 with P = trace( u) = div( u) = 0 (divergence-free flow) Q = ½ (trace( u) 2 trace(( u) 2 )) R = det( u) Δ = (Q/3) 3 + (R/2) 2 > 0 Interpretation Complex eigenvalues of u
Vortex Criteria: Q Q > 0 [Hunt et al. 1988] Eigenvalues λ 1, λ 2, λ 3 of u satisfy characteristic equation λ 3 Pλ 2 + Qλ R = 0 with Q = ½ (trace( u) 2 trace(( u) 2 )) = ½ ( Ω 2 - S 2 ) S = ½ ( u + ( u) T ) Ω = ½ ( u -( u) T ) Interpretation balance between shear strain rate and vorticity (= u) magnitude
Vortex Criteria: λ 2 λ 2 < 0 [Jeong et al. 1995] Eigenvalues λ 1 λ 2 λ 3 of S 2 + Ω 2 with S = ½ ( u + ( u) T ) Ω = ½ ( u -( u) T ) Interpretation Removes unsteady irrotational straining and viscous effects S 2 + Ω 2 represents negated Hessian of a corrected pressure p S 2 + Ω 2 = - 1/ρ Hessian(p)
Vortex Core Lines Δ > 0 and u ε r (ε r : real eigenvector of u) [Sujudi et al. 1995] Not Galilean invariant λ 2 isosurfaces vs. vortex core lines in different frames Images: Sahner et al. 2005
Vortex Core Lines Valley lines of λ 2 [Sahner et al. 2005] Galilean invariant Images: Sahner et al. 2005
Vortex Core Lines Valley lines of λ 2, topologically consistent with isosurfaces [Schafhitzel et al. 2008] Derived from isovolume skeleton Image: Schafhitzel et al. 2008
Vortex Core Lines In Unsteady Flow Generalized Sujudi to unsteady flow [Weinkauf et al. 2007] 2D: Sujudi in 3D space-time (red) by Parallel Vectors vs. crit. Pnt. tracking (blue) white: path lines in space-time 3D: Sujudi in 4D space-time by Coplanar Vectors, formulated as 3D Parallel Vectors Image: Weinkauf et al. 2007
Vortex Core Lines In Unsteady Flow Generalized Sujudi and Roth [Roth et al. 1998] to unsteady flow [Fuchs et al. 2007] Sujudi: ε r u a s u a s = ( u)u true material derivative a t = u/ t + ( u)u a t u Image: Fuchs et al. 2007
Vortex Dynamics Vorticity Transport [Sadlo et al. 2006] Images: Sadlo et al. 2006
Delocalized Quantities Delocalized vortex detectors [Fuchs et al. 2008] λ 2 < -1000 Integration of quantities along path lines Linked views + interactive masking Images: Fuchs et al. 2008 delocalized λ 2 < -5000 λ 2 < -5000
Mz An objective definition of a vortex [Haller 2005] material tubes in which material elements do not align with directions suggested by the strain eigenvectors. Based on strain acceleration tensor (Cotter-Rivlin): M = S/ t + ( S)u + S( u) + ( u) T S Z(x,t): cone in strain basis Mz: restriction of M on Z(x,t) Image: Haller 2005 Measure the time a trajectory spends in indefinite Mz If high vortex
Mz: ABC Flow λ 2 < 0 Time of indefinite Mz Images: Haller 2005
Part II: Vector Field Topology
Lagrangian Coherent Structures Vector Field Topology Lagrangian Coherent Structures (LCS) Image: Weinkauf et al. 2004 Image: Shadden et al. 2005 Crit. pts. & streamlines Instantaneous view Not Galilean invariant Ridges in Lyapunov exponent Transient view Galilean invariant
Lagrangian Coherent Structures Vector Field Topology Lagrangian Coherent Structures (LCS) Image: Weinkauf et al. 2004 Image: Shadden et al. 2005 Crit. pts. & streamlines Instantaneous view Not Galilean invariant Ridges in Lyapunov exponent Transient view Galilean invariant
LCS in Nature Confluences Glaciers from: www.publicaffairs.water.ca.gov/swp/swptoday.cfm LCS = Interfaces from: www.scienceclarified.com/ga-he/glacier.html LCS = Moraines
Finite-Time Lyapunov Exponent (FTLE) FTLE: growth of perturbation after advection time T 0 0 δ Δ 0 0, 0, 1 ln Δ/δ
FTLE Computation By pre-sampled flow map φ [Haller 2001] ϕ 0 t FTLE t 0 = Image: Shadden et al. 2005
FTLE Computation By pre-sampled flow map φ [Haller 2001] ϕ 0 t FTLE t 0 = Image: Shadden et al. 2005 ϕ 0 0 2 A 2 λ max A A
FTLE Computation By pre-sampled flow map φ [Haller 2001] ϕ 0 t FTLE t 0 = Image: Shadden et al. 2005, 0, 1 ϕ 0 0 2 A 2 λ max A A
FTLE Computation By pre-sampled flow map φ [Haller 2001] ϕ 0 t FTLE t 0 = Image: Shadden et al. 2005, 0, 1 ϕ 0 0 2 A 2 λ max A A
Efficient Computation of FTLE Garth et al. 2007 Sadlo et al. 2007 Refinement of flow map based on prediction by Catmull-Rom interpolation Refinement of FTLE based on filtered height ridge criterion Image: Garth et al. 2007 Image: Sadlo et al. 2007
Finite-Size Lyapunov Exponent (FSLE) FSLE [Aurell et al. 1997]: time needed to separate by factor s 0 0 δ sδ 0 0, 0, 1 ln s
FTLE & FSLE (Filtered) Images: Sadlo et al. 2007 FTLE T = 0.1 FSLE s= 1.5 T max = 0.1 FSLE s= 4 T max = 0.1
Stationary (Classical) 2D Vector Field Topology Image: Weinkauf et al. 2004 Critical Points Isolated zeros in vector field Classified using u Saddle: det u < 0 (yellow) Source/sink/node/focus (other) Not Galilean invariant! Separatrices [Helman et al. 1989] Streamlines converging to saddles in pos. or neg. time (white) All in terms of special streamlines Crit. pnt. = degenerate streamline
Lagrangian Coherent Structures Image: Shadden et al. 2005 Lagrangian Coherent Structures Separatrices in time-dep. fields Ridges in FTLE field May be related to special location Advect with flow (material lines) Compare: streak lines Related to special location (seed) Advect with flow LCS obtained by streak lines seeded at special locations? Generated in reverse time
Time-Dependent 2D Vector Field Topology In terms of special streak lines (generalization) [Sadlo et al. 2009] Saddle-type Critical Points Degenerate generalized streak lines Seed moves along path line (seed is a particle) Problem: which path line? Saddle-type path lines Separatrices Streak lines converging to saddle-type path lines in pos. or neg. time
Time-Dependent 2D Vector Field Topology: Hyperbolic Traj. Saddle-type path line Path line inside saddle-type (hyperbolic) region Path line along which det u < 0 Haller 2000: Finding finite-time invariant manifolds in twodimensional velocity fields Hyperbolic Trajectories (HT) Path lines converging to HT in pos. or neg. time form 2D manifolds in space-time
HT and Space-Time Manifolds of Trajectories Isotemporal slice of manifold is a generalized streak line Hyperbolic trajectory
Space-Time Streak Manifolds Space-time streak manifolds seeded along HT Seeds of space-time streak manifold (orange) Seed of hyperbolic trajectory (black)
Seeding Hyperbolic Trajectories Haller 2000: intersections of ridges ([Eberly 1996]) in forward and backward hyperbolicity times Ide et al. 2002: temporal analysis of classical critical points construction of time-dependent linear model
Seeding Hyperbolic Trajectories Haller 2000: intersections of ridges ([Eberly 1996]) in forward and backward hyperbolicity times Hyperbolicity time: time a trajectory spends in hyperbolic region (det u < 0) until it leaves Additional conditions for uniformly hyperbolic traj.
Sampling Hyperbolicity Time Direct sampling 5x5 supersampling Images: Sadlo et al. 2009
Quad Gyre Example Saddles at x=0 oscillate horizontally Saddles at x=-1 and x=1 are stationary Non-linear saddles (cosine profiles) Spatially periodic Due to Shadden et al. for FTLE vis.
Quad Gyre Hyperbolicity Time Forward hyp. time Backward hyp. time Images: Sadlo et al. 2009
Quad Gyre Hyperbolicity Time Forward hyp. time Backward hyp. time Images: Sadlo et al. 2009
Quad Gyre Uniform Hyperbolicity Forward uniform hyp. Backward uniform hyp. Images: Sadlo et al. 2009
Quad Gyre Uniform Hyperbolicity Forward uniform hyp. Backward uniform hyp. Images: Sadlo et al. 2009
Quad Gyre FTLE Forward FTLE Backward FTLE Images: Sadlo et al. 2009
Quad Gyre Space-Time Streak Manifolds Image: Sadlo et al. 2009
Quad Gyre Space-Time Streak Manifolds Image: Sadlo et al. 2009
Seeds for Hyperbol. Trajectories: Scale-Dependent Analysis FTLE with T=0.5 FTLE with T=3 Images: Sadlo et al. 2009
Buoyant Flow CFD Example Image: Sadlo et al. 2009
Summary Vortices Criteria Core lines Dynamics Lagrangian coherent structures FTLE FSLE Space-time streak manifolds
References [Aurell et al. 1997] E. Aurell, G. Boffetta, A. Crisanti, G. Paladin, A. Vulpiani. Predictability in the large: an extension of the concept of lyapunov exponent. J. Phys. A: Math. Gen, 30:1.26, 1997. [Chong et al. 1990] M. S. Chong, A. E. Perry, B. J. Cantwell. A general classification of three-dimensional flow field. Phys. Fluids A 2, 765, 1990. [Eberly 1996] D. Eberly. Ridges in Image and Data Analysis. Computational Imaging and Vision. Kluwer Academic Publishers, 1996. [Fuchs et al. 2007] R. Fuchs, R. Peikert, H. Hauser, F. Sadlo, P. Muigg. Parallel Vectors Criteria for Unsteady Flow Vortices. IEEE Transactions on Visualization and Computer Graphics, Vol. 14, No. 3, pp. 615-626, 2008. [Fuchs et al. 2008] R. Fuchs, R. Peikert, F. Sadlo, B. Alsallakh, E. Gröller. Delocalized Unsteady Vortex Region Detectors. Proceedings VMV 2008, pp. 81-90, 2008.
References [Garth et al. 2007] C. Garth, F. Gerhardt, X. Tricoche, H. Hagen. Efficient Computation and Visualization of Coherent Structures in Fluid Flow Applications, In Proceeding of IEEE Visualization 2007, pp. 1464-1471, 2007. [Hunt et al. 1988] J. C. R. Hunt. Vorticity and vortex dynamics in complex turbulent flows. In Proc. CANCAM, Trans. Can. SOC. Mech. Engrs 11, 21. 1987. [Haller 2000] G. Haller. Finding finite-time invariant manifolds in two-dimensional velocity fields. Chaos 10, 99 108, 2000. [Haller 2001] G. Haller. Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149, 248 277, 2001. [Haller 2005] G. Haller. An objective definition of a vortex. Journal of Fluid Mechanics, 525:1 26, 2005.
References [Helman et al. 1989] J. Helman, L. Hesselink. Representation and display of vector field topology in fluid flow data sets. Computer 22, 8, 27 36, 1989. [Jeong et al. 1995] J. Jeong, F. Hussain. On the identification of a vortex. Journal of Fluid Mechanics, 285:69 84, 1995. [Robinson 1991] S. K. Robinson. Coherent Motions in the Turbulent Boundary Layer. Ann. Rev. Fluid Mechanics, 23:601-639, 1991. [Roth et al. 1998] M. Roth, R. Peikert. A higher-order method for finding vortex core lines. In Proceedings IEEE Visualization 1998, 143 150, 1998. [Sadlo et al. 2006] F. Sadlo, R. Peikert, M. Sick. Visualization Tools for Vorticity Transport Analysis in Incompressible Flow. IEEE Transactions on Visualization and Computer Graphics, Vol. 12, No. 5, pp. 949-956 2006.
References [Sadlo et al. 2007] F. Sadlo, R. Peikert. Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction. IEEE Transactions on Visualization and Computer Graphics, Vol. 13, No. 6, pp. 1456-1463, 2007. [Sadlo et al. 2009] F. Sadlo, D. Weiskopf. Time-Dependent 2D Vector Field Topology: An Approach Inspired by Lagrangian Coherent Structures. Computer Graphics Forum, accepted for publication, 2009. [Sahner et al. 2005] J. Sahner, T. Weinkauf, H. C. Hege. Galilean Invariant Extraction and Iconic Representation of Vortex Core Lines. In EuroVis 2005, 151-160, 2005. [Shadden et al. 2005] S. C. Shadden. Lagrangian coherent structures. http://www.cds.caltech.edu/ shawn/lcs-tutorial/contents.html, 2005.
References [Schafhitzel et al. 2008] T. Schafhitzel, J. Vollrath, J. Gois, D. Weiskopf, A. Castelo, T. Ertl. Topology-Preserving lambda2-based Vortex Core Line Detection for Flow Visualization. Computer Graphics Forum (Eurovis 2008), 27(3):1023-1030, 2008. [Sujudi et al. 1995] D. Sujudi and R. Haimes. Identification of swirling flow in 3D vector fields. Technical Report AIAA-95-1715, American Institute of Aeronautics and Astronautics, 1995. [Weinkauf et al. 2007] T. Weinkauf, J. Sahner, H. Theisel, H.-C. Hege. Cores of Swirling Particle Motion in Unsteady Flows. IEEE Transactions on Visualization and Computer Graphics, 13(6), 2007.